Alma Mater Studiorum · Università di Bologna Friedrich-Alexander-Universität Erlangen-Nürnberg DOTTORATO DI RICERCA IN MATEMATICA Ciclo XXIV Settore Concorsuale di afferenza: 01/A4 Settore Scientifico disciplinare: MAT/07 RESONANCES IN THE TWO CENTERS COULOMB SYSTEM Presentata da: MARCELLO SERI Coordinatore Dottorato: ALBERTO PARMEGGIANI Relatori: MIRKO DEGLI ESPOSTI ANDREAS KNAUF Esame finale anno 2012 Acknowledgements I would like to use this space to thank the people that directly or indirectly contributed to this thesis. First of all, I would like to thank Andreas Knauf and Mirko Degli Esposti for their incredible support and the trust they have always placed in me. Not to mention all the time that they spent with me or for me, and the enthusiasm that they have always been able to transmit to me. There are no words to thank them enough. Secondly, I would like to thank Thierry Jecko for all his help. My stay in Cergy has been really enjoyable and the time spent together has been essential for the realization of this thesis. How not to mention my friends and coworkers Sylvain Golénia, Marco Falconi, Max Lein, Giampaolo Cristadoro, Marco Lenci, Giuseppe De Nittis, Peter Gmeiner, and Maxim Drabkin. I want to thank them for all the great moments spent together. There are other people who deserve a special mention. I am very thankful to Hermann Schulz-Baldes for many interesting discussions and suggestions. And I must thank Frau Moch, without her I would probably have been lost in a huge pile of documents. Even if somebody may look surprised, I have a non-scientific life too. I want to thank my family and my friends. They know that I will owe a debt of gratitude to them that cannot be filled. Their support have been essential to me and the time that we spent together is invaluable. Thanks! 3 Contents Sunto 7 Zusammenfassung 9 Chapter 1. Introduction 1.1. The two-center Coulomb system 1.2. Separation in prolate elliptic coordinates 11 13 14 Chapter 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2. The classical problem of two Coulomb centers Hamiltonian setting Bifurcation diagrams Bifurcations for planar motions Motion for Z− = 0 Motion for Z+ = 0 The general case 19 19 23 24 28 31 32 Chapter 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3. The two-centers system on L2 (R3 ) First separation from an operator theoretical point of view A remark on the second separation Self-adjoint extension and spectrum of the angular operator Analytic extensions of the eigenvalues of Aηλ Second separation from an operator theoretical point of view Self-adjoint extensions of the radial operators A lλ Formal ”partial wave expansion” of the Green’s function 35 35 37 38 41 42 43 47 Chapter 4.1. 4.2. 4.3. 4. The two-centers system on L2 (R2 ) Separation of variables Spectrum of the angular equation and its analytic continuation Asymptotic behavior of solution of the radial Schrödinger equation and their analytic extensions Generalized eigenfunctions, Green’s function and the scattering matrix Formal ”partial wave expansion” of the Green’s function Resonances for the 2-dimensional problem Computation of the resonances of Kξ Eigenvalues asymptotics and resonant regions for Z− = 0 near the bottom of the spectrum Eigenvalues asymptotics and resonant regions for Z− > 0 near the bottom of the spectrum 49 49 51 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 5 54 66 69 71 72 74 75 6 CONTENTS 4.10. High energy estimates 4.11. Numerical investigations Chapter 5.1. 5.2. 5.3. 5. Back to the three dimensional problem Asymptotic analysis of the solutions of A lλ Estimate for the eigenvalues in the case Z− = 0 Estimate for the eigenvalues in the case Z− ̸= 0 Chapter 6. Final considerations 80 81 89 89 91 95 99 Appendix A. A small introduction to the theory of Sturm-Liouville operators 101 Appendix B. Singular second-order Sturm-Liouville operators with complex coefficients 105 Appendix C. Perturbation theory in a nutshell 111 Appendix D. Generalized Prüfer transformation in the semi-classical limit 115 Bibliography 121 Sunto La tesi è incentrata sulla comprensione delle risonanze quantistiche per il sistema coulombiano a due centri fissi in due e tre dimensioni. L’operatore di Schrödinger semiclassico − h2 2 ∆− Z1 |q − s1 | − Z2 |q − s2 | in L2 (R2 ) (o in L2 (R3 )) descrive la funzione d’onda di una particella in un campo generato da due cariche coulombiane fisse Z1 e Z2 nello spazio bi- e tridimensionale. Le cariche sono considerate con valori arbitrari. Seppure semplificato, il problema coulombiano a due centri fissi rappresenta un modello per lo studio di ioni molecolari con un elettrone (ad esempio l’idrogeno molecolare ionizzato H2+ ). Il problema, che risale alla fine del 1800, è stato studiato classicamente da Jacobi in meccanica celeste e introdotto come modello quantistico da Pauli nella sua tesi di dottorato nel 1922. La letteratura è vasta e si protrae fino a questo decennio ma riguarda quasi esclusivamente l’analisi del problema classico e quantistico per energie negative. Il caso dei due centri è particolarmente interessante in quanto rappresenta il modello molecolare non banale più semplice: la dinamica classica è integrabile. In particolare, tramite le coordinate prolate ellittiche è possibile ricondursi allo studio di un sistema di equazioni di Sturm-Liouville accoppiate a cui è possibile applicare solide tecniche di analisi funzionale e teoria semiclassica. Nella prima parte del lavoro cerchiamo di completare il quadro classico descrivendo il diagramma delle biforcazioni e la struttura del ritratto di fase per energie non negative. Oltre ad essere interessanti di per se, queste sono importanti anche per comprendere se e dove è possibile aspettarsi la presenza di risonanze quantistiche. Successivamente ci concentriamo sullo studio del problema quantistico, sviluppando gli strumenti e le costruzioni necessari per poter dare una definizione di risonanze e per poter implementare le stime per la loro localizzazione. Le risonanze corrispondono a stati meta-stabili microlocalizzati nello spazio delle fasi sulla varietà (in)stabile dell’orbita classica che oscilla tra i due centri. In quersto lavoro le risonanze sono definite come autovalori complessi generalizzati con parte immaginaria piccola. Tramite diverse tecniche di approssimazione le risonanze sono enumerate in termini di 2 o 3 parametri (per il problema in R2 e in R3 rispettivamente) e localizzate in modo esplicito con un errore dell’ordine di 7 8 SUNTO 3 h 2 (eventualmente migliorabile usando una espansione asintotica) e in diversi regimi asintotici dei parametri. Rispetto al caso bidimensionale, in cui le risonanze descrivono un reticolo ben separato nella fascia del piano complesso vicina all’asse reale, il caso tridimensionale presenta un’ulteriore difficoltà data dalla possibile presenza di valori degeneri. Zusammenfassung In dieser Doktorarbeit wurden die Resonanzen für das quantenmechanische Zweizentren-Problem in zwei und drei Dimensionen studiert. Der semiklassische Schrödinger-Operator, definiert durch − h2 2 ∆− Z1 |q − s1 | − Z2 |q − s2 | auf L2 (R2 ) (oder auf L2 (R3 )), beschreibt die Dynamik der Wellenfunktion eines Teilchens in einem Feld, das durch zwei Coulomb-Zentren beliebiger elektronischer Ladungen Z1 und Z2 erzeugt ist. Obwohl vereinfacht, ist das Zweizentren-Problem ein realistisches Modell zur Untersuchung der molekularen Ionen mit einem Elektronenstrahl (z.B. das ionisierte Wasserstoffmolekül H2+ ). Das klassische Problem ist von Jacobi am Ende des neunzehnten Jahrhunderts in der Himmelsmechanik studiert werden, und das quantenmechanische Modell hat Pauli in seiner Dissertation im Jahr 1922 eingeführt. Die Literatur ist groß, mit Veröffentlichungen bis in den letzten Jahren, aber sie behandelt ausschließlich die klassischen und quantenmechanische Analyse des Problems für negative Energien. Das Zweizentren-Problem ist besonders interessant, denn es stellt das einfachste nicht triviale molekulare Modell dar, denn die klassischen Dynamik ist integrierbar. Insbesondere kann das Modell anhand der prolat elliptischen Koordinaten auf ein System von Sturm-Liouville-Gleichungen reduziert werden. Und für dieses System kann man starke Techniken aus Funktionalanalysis und semiklassischer Theorie anwenden. Im ersten Teil der Arbeit vervollständigen wir das klassische Bild mit dem Verzweigungsdiagramm und dem Phasenportrait für positive Energien. Das ist für sich genommen interessant, aber es ist auch wichtig zu verstehen, ob quantenmechanische Resonanzen existieren, und wo sie sich befinden. Dann konzentrieren wir uns auf die Untersuchung des quantenmechanischen Modells. In diesem Teil entwickeln wir die Techniken und die Strukturen, um eine Definition der Resonanzen bereitzustellen und die Abschätzungen für ihre Lokalisierung auszuführen. Die Resonanzen entsprechen metastabilen Zuständen, die im Phasenraum auf der (in-)stabilen Mannigfaltigkeit des zwischen den Zentren oszillierenden geschlossenen Orbits mikrolokalisiert sind. 9 10 ZUSAMMENFASSUNG Die Resonanzen werden als verallgemeinerte komplexe Eigenwerte mit kleinen Imaginärteilen definiert. Sie werden durch verschiedene Approximationstechniken aufgezählt und in verschiedenen asymptotischen Regimen von Parametern mit einem 3 Fehler der Ordnung h 2 explizit lokalisiert. In die zweidimensionalen Fall beschreiben die Resonanzen ein reguläres Gitter im Bereich der komplexen Ebene in der Nähe der reellen Achse. Andererseits hat der dreidimensionale Fall eine schwierigere Struktur und möglicherweise degenerierte Resonanzen. CHAPTER 1 Introduction Our work mainly concerns the study of the quantum mechanical two fixed center Coulomb system in two and three dimensions. The setting will be defined in the next sections, but before entering into details we would like to give an overview on the problem and his history and describe the structure of the thesis. The classical motion of a test particle in the field of two fixed centers was first considered by Euler in 1760 and it represents one of the most famous integrable problem of classical mechanics [Kna11]. It was Jacobi that in 1884 proved its integrability and gave explicit solutions of the equations of motion in terms of elliptic functions [Jac84]. The first development of the corresponding quantum mechanical model is due to Pauli that, in 1922, even before the Schrödinger formulation of quantum mechanics, applied it to the semi-classical hydrogen molecular ion H2+ [Pau22]. Ten years after Pauli’s dissertation, the Schrödinger equation was applied to the problem by Jaffé [Jaf34] and Baber and Hassé [BH35] that described the system in its modern form. Starting from this moment, the two-center problem is fully considered in molecular physics as the simplest model for one-electron diatomic molecules. There has always been a deep connection between classical mechanics and quantum mechanics. The understanding of the classical picture is essential for the comprehension of the semi-classical and the quantum mechanical ones. For classical coulombic problems (including the two-centers one) the literature goes from Jacobi up to the recent years with the work of [WDR04] that gives a complete picture of the phase-space structure for negative energies and [KK92, Kna02] that gives a geometrical description of the classical scattering. Like for the classical model, there is a huge amount of literature related to the quantum counterpart. Despite this fact, the understanding of the system is far from being complete and a lot is known only for what concerns the discrete spectrum (i.e. negative energies) or in some special limits in which the two-center potential converges to a one-center one [Har80, GGHS85, BHL06, LKJ04, SAFG06]. Of major interest are the works of Strandt and Reinhardt [SR79] that gave an excellent semi-classical description of the model for the hydrogen ion and of its boundstates and [Cha90] that studied the low-lying eigenvalues for the semi-classical two-dimensional problem. The results related to the quantum or semi-classical scattering are really few and the most important advances can be found in [Lea86, GG76] and [Liu92]1. In fact most of the results in the literature are related to the hard problem of finding an 1Notice that in this paper the computation of the Green’s function is known to be wrong. 11 12 1. INTRODUCTION algorithm to obtain numerical approximations of the scattering solutions. In contrast, nothing or really few is known on the regularity of the solutions with respect to the parameters of the system and the problem of resonances. Quantum resonances are a key notion of quantum physics: roughly speaking they are scattering states (i.e. states of the essential spectrum) that for long time behave like bound states (i.e. eigenfunctions). They are usually defined as poles of a meromorphic function, but there is not really a unique way to study and define them [Zwo99]. On the other hand, it is known that their many definitions coincide in some settings [HM87] and that their existence is related to the presence of some classical orbits “trapped” by the potential. If a quantum systems has a potential presenting a positive local minimum above its upper limit at infinity, for example, it is usually possible to find quantum resonances, called shape resonances. They are related to the classical bounded trajectories around the local minimum [HS96]. They are not the only possible ones, in fact it has been proved in [BCD87b, BCD88, Sjö87, GS87] that there can be resonances generated by closed hyperbolic trajectories or by a non-degenerate maximum of the potential. The main difference is that the shape resonances appear to be localized much nearer to the real axis with respect to this last one. Even the presence or absence of the resonances is strictly related to the classical picture. In fact it is possible to use some classical estimates, called nontrapping conditions, to prove the existence of resonance free regions (see for example [BCD87a, Mar02, Mar07]). In this work we want to investigate the existence of resonances for two-centers Coulomb systems with arbitrary charges, defining them in terms of generalized complex eigenvalues of a non-selfadjoint deformation of the two-center Schrödinger operator. The fact that the non-trapping condition does not hold for the two centers problem [CJK08], and the presence of a single bounded hyperbolic orbit for big positive energies (see [KK92, Theorem 6.9] and [Kna02, Theorem 12.8]) suggests that resonances are likely to be present. In view of the last two theorems is even clear that the two-centers system is a good model to perform this study, furthermore it is known there that in presence of more coulombic singularities the system is no more integrable and there is a Cantor set of hyperbolic trapped orbits [KK92, Kna02]. The thesis is structured as follows. The rest of this chapter is used to define the three dimensional two-centers Coulomb system and show how its differential equation separates in prolate elliptic coordinates. In Chapter 2 we integrate the description of the classical bifurcation diagram initiated by [WDR04]. Namely we add the bifurcation diagram of the planar twocenters system for positive energies and identify the "trapped" trajectories. In Chapter 3 we analyze the separation of the operator in R3 from an operator theoretical point of view and we clarify formally how we want to define the resonances and the main problems arising. In Chapter 4 we concentrate on the two-centers problem in R2 . In the first part we analyze the separation of the problem in its radial and an angular part. We then perform an analytic continuation of the eigenvalues of the angular part in 1.1. THE TWO-CENTER COULOMB SYSTEM 13 the complex plane and we study the asymptotic form of the radial solutions and their analyticity with respect to the energy parameter. With these we define the resolvent kernel (the Green’s function), the generalized eigenfunctions (scattering states) and the scattering matrix and show that all these objects admit an analytic continuation in the energy parameter suitable to define the resonances of the system. Finally we compute some approximations of the resonances and perform a numerical investigation of their structure. In Chapter 5 we relate the results of Chapter 4 and of a paper of Agmon and Klein [AK92] to describe the resolvent kernel (the Green’s function), the generalized eigenfunctions (the scattering states) and the scattering matrix for the operator in three dimensions and define its resonances. In Chapter 6 we summarize the results of the thesis and describe the main open problems and further direction of work. The main theories and tools needed for the work are described and referenced in the appendices. 1.1. The two-center Coulomb system We consider the operator in L2 (R3 ), given for h > 0 by H := −h 2 ∆ + V (q) with V (q) := −Z1 |q − s1 | + −Z2 |q − s2 | . (1.1.1) This describes the motion of an electron in the field of two nuclei of charges Zi ∈ R∗ := R \ {0}, fixed at positions s1 ̸= s2 ∈ R3 , taking into account only the electrostatic force. By the unitary realization U f (x) := | det A|−1/2 f (Ax + b ) of an affinity of Rd we assume that the two centers are at s1 := a := 1 0 0 and s2 := −a . Remark 1.1. For Z1 = Z2 = 0 or in case only one of the charges is zero, the problem would reduce respectively to the free Laplacian or the one-center problem. In particular it would not be a two centers problem anymore. Therefore the restriction Zi ∈ R∗ . Remark 1.2. Notice that if we set Z1 = Z2 = 1 in the operator in (1.1.1), we get the Schrödinger operator for the simply ionized hydrogen molecule H2+ [SR79]. In the same way, if Z1 = 1 = −Z2 , it describes an electron moving in the field of a proton and an anti-proton [GGHS85]. Another important example covered by the operator is the doubly charged heliumhydride molecular ion H e H ++ , in this case Z1 and Z2 are both positive but of different magnitude [WDR04]. Remark 1.3. Even if (1.1.1) doesn’t directly describe the diatomic molecular scattering, it is strictly related to it: the molecular scattering is generally studied by the Thomas Fermi approximation [LS77]. The resulting equation appears as a perturbation of a purely Coulombic potential V by a smooth term W . The Coulombic potential then represents the heavy atoms as fixed while the smooth perturbation represents the cloud of electrons of the molecule. 14 1. INTRODUCTION Remark 1.4. In this thesis we use mathematicians’ notation h for what the physicists call ħh . It means that for us h is generally a small parameter for asymptotic analysis and it is not necessarily interpreted as the Planck constant. 1.2. Separation in prolate elliptic coordinates We define M := R+ × (0, π) × [0, 2π). The restriction to M of the map from ξ cosh(ξ ) cos(η) η ∈ M 7→ sinh(ξ ) sin(η) cos(ϕ) ϕ sinh(ξ ) sin(η) sin(ϕ) ∈ R3 defines a C ∞ diffeomorphism G : M → G(M ) (1.2.1) whose image G(M ) = R3 \ (R × {0} × {0}) is dense in R3 , therefore it defines a change of coordinate from q ∈ R3 to (ξ , η, ϕ) ∈ M . The new coordinates (ξ , η, ϕ) are called prolate elliptic coordinates. q2 1 -1 1 q1 -1 Figure 1.1. Prolate elliptic coordinate for ϕ ∈ {0, π}. Remark 1.5. The curve ϕ = c describes a half plane, in particular ϕ = 0 (ϕ = π) is the (q1 , q2 )-half-plane with q2 > 0 (q2 < 0). In the (q1 , q2 )-plane the curves ξ = c are half ellipses with foci at ±a while the curves η = c are confocal half hyperbolas. Remark 1.6. The Jacobian determinant det(DG(ξ , η, ϕ)) = sin(η) sinh(ξ )(sin2 (η) + sinh2 (ξ )), thus the change of coordinates is degenerate for η ∈ {0, π} or ξ = 0. This happens because the q1 axes is pointwise invariant under rotations of angle ϕ, in fact for the values ξ = 0 the η coordinate would parametrize the q1 -axis interval between the two centers, and for η = 0 (η = π) the ξ coordinate would parametrize the positive (negative) q1 -axis with |q1 | > 1. The importance of this change of coordinate is clarified by the following theorem (see e.g. [BH35]). 1.2. SEPARATION IN PROLATE ELLIPTIC COORDINATES 15 Theorem 1.7. Let ¦ © u ∈ Ca (R3 ) := C (R3 ) | u↾R3 \{±a} is twice continuously differentiable . The eigenvalue equation −h 2 ∆ + V (q) u(q) = E u(q), E ∈ R, transformed to prolate elliptic coordinates, separates with the ansatz u ◦ G(ξ , η, ϕ) = f (ξ ) g (η)e i λϕ into the decoupled system of ordinary differential equations 2 h 2 λ2 2 2 2 − Z cosh(ξ ) − E cosh (ξ ) + µ f (ξ ) = 0 −h ∂ − h coth(ξ ) ∂ + + ξ ξ sinh2 (ξ ) h 2 λ2 2 2 2 2 −h ∂η − h cot(η) ∂η + sin2 (η) + Z− cos(η) + E cos (η) − µ g (η) = 0 −∂ϕ2 − λ2 e i λϕ = 0. Here λ ∈ Z, µ ∈ C and ¦ © f ∈ CN2 (R0+ ) := h ∈ C 2 (R0+ ) | h ′ (0) = 0 , ¦ © 2 g ∈ Cper ([0, π]) := h ∈ C 2 ([0, π]) | h (k) (0) = h (k) (π) for k = 0, 1, 2 , ϕ ∈ [0, 2π) and we have set Z± := Z2 ± Z1 and ∂α := ∂ . ∂α Remark 1.8. Without restriction we may assume Z− ∈ R0+ and Z+ ∈ R, Z+ ̸= Z− , i.e. Z2 ≥ Z1 . We set r1 := |q − s1 |, r2 := |q − s2 | and transform to prolate elliptic coordinates. We have Proof of 1.7. 2 r2,1 = (q1 ± 1)2 + q22 + q32 = (cosh(ξ ) cos(η) ± 1)2 + sinh2 (ξ ) sin2 (η) cos2 (ϕ) + sin2 (ϕ) = cosh2 (ξ ) cos2 (η) + sin2 (η) ± 2 cosh(ξ ) cos(η) + (1 − sin2 (η)) = (cosh(ξ ) ± cos(η))2 . Thus the distances from the centers equal r1 = cosh ξ − cos η and r2 = cosh ξ + cos η. Defined the function F by F (ξ , η) := sinh2 (ξ ) + sin2 (η) = cosh2 (ξ ) − cos2 (η), (1.2.2) we obtain V (ξ , η, ϕ) = − =− Z1 − Z2 =− 1 Z+ (r2 + r1 ) + Z− (r1 − r2 ) r1 r2 2 Z+ cosh(ξ ) − Z− cos(η) F (ξ , η) r1 r2 , (1.2.3) 16 1. INTRODUCTION and the Laplacian is given by ∆= 1 F (ξ , η) ( ∂ξ2 + coth(ξ ) ∂ξ + ∂η2 + cot(η) ∂η + + = 1 ¨ F (ξ , η) 1 sin2 (η) 1 sinh(ξ ) + ∂ϕ2 sinh2 (ξ ) ∂ξ sinh(ξ ) ∂ξ + 1 2 + ) ! 1 sin (η) ! 1 + 2 sinh (ξ ) 1 sin(η) ) ∂η sin(η) ∂η + ∂ϕ2 . (1.2.4) Let u ∈ Ca (R3 ). We first consider the ansatz u ◦ G(ξ , η, ϕ) = ue(ξ , η)e i λϕ where to preserve the continuity and differentiability of the u we have to assume n ue ∈ h ∈ C 2 (R+ × [0, π]) | h is η-periodic, o ∂ξ h(ξ , η)|ξ =0 = 0 for η ∈ (0, π) and ϕ ∈ [0, 2π). The eigenvalue equation −h 2 ∆ − V ◦ G u ◦ G = E u ◦ G separates in the form h 2 ∂ (sinh(ξ ) ∂ ) −h 2 ξ sinh(ξ ) ξ 2 2 h ∂η sin(η) ∂η −h sin(η) −∂ϕ2 − λ2 e i λϕ = 0 + h 2 λ2 sin2 (η) + h 2 λ2 sinh2 (ξ ) e + V (ξ , η) ue = 0 where Ve (ξ , η) := −Z+ cosh(ξ ) + Z− cos(η) − E cosh2 (ξ ) + E cos2 (η). Then, with the ansatz ue(ξ , η) = f (ξ ) g (η) with 2 f ∈ CN2 (R+ ) and g ∈ Cper ([0, π]) the first equation separates and we obtain the decoupled system of ordinary differential equations 1 2 λ −h ∂ sinh(ξ ) ∂ f (ξ ) + V (ξ ) + µ f (ξ ) = 0 ξ ξ sinh(ξ ) ξ 1 ∂η sin(η) ∂η g (η) + Vηλ (η) − µ g (η) = 0 −h 2 sin(η) −∂ϕ2 − λ2 e i λϕ = 0 (1.2.5) 1.2. SEPARATION IN PROLATE ELLIPTIC COORDINATES 17 where Vξλ (ξ ) := −Z+ cosh(ξ ) − E cosh2 (ξ ) + Vηλ (η) := Z− cos(η) + E cos2 (η) + h 2 λ2 sinh2 (ξ ) h 2 λ2 sin2 (η) . , (1.2.6a) (1.2.6b) □ Remark 1.9. Here the separation constant µ plays the role of the spectral parameter in time independent Schrödinger equations, and the original energy E plays the role of a coupling constant. CHAPTER 2 The classical problem of two Coulomb centers In this section we want to extend the results of [WDR04] and the discussions of [Kna02, Kna11] to the case of positive energies and general values of the charges. 2.1. Hamiltonian setting We consider the classical Hamiltonian function on the cotagent bundle T ∗ (R3 \ {±a}) relative to the 2-center potential of (1.1.1), where we have alerady rescaled the two centers to be in ±a : H ( p, q) := − | p|2 2 + −Z1 |q − a| + −Z2 |q + a| . Theorem 2.1. The Hamiltonian H ( p, q) restricted to T ∗ (R3 \ (R, 0, 0)) is transformed by the prolate elliptic coordinates into the Hamiltonian H ( pξ , pη , pϕ , ξ , η, ϕ) := 1 F (ξ , η) (H1 ( pξ , pϕ , ξ ) + H2 ( pη , pϕ , η)) (2.1.1) where F is defined in (1.2.2) and H1 ( pξ , pϕ , ξ ) := H2 ( pη , pϕ , η) := pξ2 2 pη2 2 + + pϕ2 2 sinh2 (ξ ) pϕ2 2 sin2 (η) − Z+ cosh(ξ ), (2.1.2) + Z− cos(η). There are three independent constants of motion H , L := H1 − cosh2 (ξ )H and pϕ with values E , K and l respectively. Proof. If we apply to H ( p, q) the canonical transformation induced by the prolate elliptic coordinates (1.2.1), the potential is transformed as in (1.2.3) and the impulses p = ( p1 , p2 , p3 ) are transformed according to pξ p 1 p = (DG(ξ , η, ϕ)−1 ) t pη . 2 p3 pϕ From (DG(ξ , η, ϕ)−1 )(DG(ξ , η, ϕ)−1 ) t = (DG(ξ , η, ϕ) t DG(ξ , η, ϕ))−1 and 0 0 F (ξ , η) t (DG(ξ , η, ϕ) DG(ξ , η, ϕ)) = 0 F (ξ , η) 0 0 0 sinh2 (ξ ) sin2 (η) we get that the Hamiltonian is transformed into (2.1.1). 19 20 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS The angular momentum pϕ around the q1 -axis is a constant of motion, in fact pϕ = q2 p3 − q3 p2 and for a solution of the Hamilton’s equations d dt pϕ = q̇2 p3 − q̇3 p2 + q2 ṗ3 − q3 ṗ2 = p2 p3 − p3 p2 + (q2 q3 − q3 q2 )r ∂W ∂r (q1 , r ) = 0. Here we have defined q r := q22 + q32 and W (q1 , r ) := Æ −Z1 −Z2 +Æ . 2 2 2 2 (q1 − 1) + r (q1 + 1) + r Given an initial condition x0 ∈ T ∗ (R3 \ {±a}) we set l := pϕ (x0 ) and E := H (x0 ). Equation (2.1.1) can be again separated [Kna11, Lemma 10.38] passing to the extended phase space and using a new time parameter s defined by dt ds = F (ξ , η). We obtain the new hamiltonian e := F (ξ , η)(H − E) = H + H H ξ η where Hξ := H1 − cosh2 (ξ )E and Hη := H2 + cos2 (η)E, (2.1.3) e −1 (0), H e describes the time evolution of H −1 (E) up to then on the submanifold H a time reparametrization. Therefore have a third constant of motion other than H and pϕ : L := H1 − cosh2 (ξ )H = −(H2 + cos2 (η)H ). (2.1.4) Setting K := Hξ (x0 ) = −Hη (x0 ) we have three constants of motion H , Hξ and pϕ whose values are denoted respectively E , K and l . Notice that these functions are generally independent in the following sense. Being real analytic functions, the subset of phase space where independence is violated, is of Lebesgue measure zero. □ Remark 2.2. By the separation of ϕ and the constancy of pϕ = l we can restrict the analysis for a given l to a reduced phase space T ∗ (R2 \ {±a}) with Hamiltonian b ( p , p , ξ , η) = H ( p , ξ ) + H ( p , η), H l ξ η ξ ξ η η where the Hξ and Hη are the one defined in (2.1.3) and have the form Hξ ( pξ , ξ ) := pξ2 2 + Vξ (ξ ), Hη ( pη , η) := pη2 2 + Vη (η). (2.1.5) Here Vξ and Vη are formally the one defined by (1.2.6) with λ = l and h = 1. If l ̸= 0 the effective potential Vξ (ξ ) (Vη (η)) becomes infinite as ξ → 0 (η → 0, η → π) thus the trajectory can never leave R3 \ (R × {0} × {0}). Therefore on 2.1. HAMILTONIAN SETTING 21 T ∗ (R3 \ (R × {0} × {0})), the Hamiltonian H generates a complete vector field and the flow exists for all times. It will be computationally useful for what follows to introduce a new coordinate change. The restriction to M 2 := (1, ∞) × (−1, 1) of the map x y 2 ∈ M 7→ arccosh(x) arccos(y) ∈ R2 defines a C ∞ diffeomorphism G 2 : M 1 → G 2 (M 2 ) (2.1.6) with image G 2 (M 2 ) = R+ × (0, π), therefore it defines a change of coordinates from (ξ , η) ∈ R+ × (0, π) to (x, y) ∈ M 2 . Theorem 2.3. The diffeomorphism defined in (2.1.6) induces an hamiltonian b 2 : T ∗ (R × (0, π)) → T ∗ M 2 . symplectomorphism G + Proof. It is enough to choose the generating function px S2 (ξ , η, p x , py ) := (cosh(ξ ), cos(η)) . py It induces a canonical transformation [Kna11, Chapter 10.5] b 2 : ( p , p , ξ , η) 7→ ( p , p , x, y) G ξ η x y where x y pξ pη = ∂ S2 ∂ px = ∂ S2 ∂ξ ∂ S2 ∂ py ∂ S2 ∂η = cosh(ξ ) , cos(η) sinh(ξ ) p x = sin(η) py □ Remark 2.4. The change of variable defined in the previous theorem corresponds to the change of variables defined by the restriction to M 3 := (1, ∞) × (−1, 1) × [0, 2π) of xy Æ x 3 3 2 G : y (x − 1)(1 − y 2 ) cos(ϕ) ∈ M 7→ Æ 2 ϕ (x − 1)(1 − y 2 ) sin(ϕ) ∈ R3 . The Jacobian determinant det(DG 3 (x, y, ϕ)) = x 2 − y 2 is singular only at the two centers (x, y) = (1, ±1), thus we can extend it to the whole M 3 \ (1, ±1, [0, 2π)). 22 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS If l = 0 the motion is effectively planar, thus we can restrict it to be in the (q1 , q2 )-plane. In this case the trajectories are allowed to pass through the q1 -axis (where the prolate elliptic coordinate are singular) and even collide with the two centers at ±a = (±1, 0). There are some different ways to regularize the motion both in the planar and spatial cases (and with an arbitrary number of centers), we refer the reader to [Kna02, Chapter 4-5], [KK92, Chapter 3] and [Kna11, Remark 11.24] for more details and references. We are going to use the regularization scheme of [KK92] for the planar two-center case, extended to the case of repelling centers. Let O := R2 and Ô := R2 \ {±a} be the configuration space, let RE := {q ∈ Ô | V (q) ≥ E} be the complement of the Hill’s region for a given energy E and let ÔE := Ô \RE be the Hill’s region for a given energy E . Notice that for E > 0 the set RE is bounded, and it is nonempty in case there is a repelling singularity (Z# < 0). Denoted ĝ the restriction to Ô of the euclidean metric on R2 , we introduce the Jacobi metric ĝE : ĝE (q) := 1 − V (q) E ĝ (q), q ∈ Ô, (2.1.7) where E is the energy of the particle whose motion we want to analyze. The solutions of the geodesic motion with unit velocity on (ÔE , ĝE ) coincide, up to a time reparametrization, with the flow on the configuration space ΦEt generated by the Hamiltonian H for initial conditions with energy E [Kna11, Section 8.5]. On the other hand (M̂ E , ĝE ) is not geodesically complete. Therefore we identify q ∈ O with q ∈ C and consider the Riemann surface ¦ © O := (q, Q) ∈ C × C | Q 2 = (q − a)(q + a) (2.1.8) as branched covering surface for O , in fact if π : O = O is the projection on the first coordinate π(q, Q) → q , it is a two-sheeted branched covering whose branch points ±a := (±a, 0) are of order one and project on the positions ±a of the nuclei. Denoting ÔE := π−1 (ÔE ), the restriction of π to ÔE leads to a two-fold unbranched covering π̂E : ÔE → ÔE and we can lift the Jacobi metric ĝE to the metric ĝE := π̂∗E ĝE on ÔE . In this way we obtain an incomplete geodesic motion on the unit tangent bundle S ÔE := {X ∈ T ÔE | ĝE (X , X ) = 1} that can be related to the restriction of the flow of the Hamiltonian restricted to the energy shell ŜE := {( p, q) ∈ T ∗ ÔE | H ( p, q) = E} by a two-sheeted unbranched covering π̃E : S ÔE → ŜE (q̇, q) 7→ p 2E(1 − V (π(q))/E) Tq π(q̇), π(q) . Theorem . For Z± > 0, ĝE extends uniquely to a smooth metric gE on 2.5 OE := π−1 Ô E defining a complete Riemann surface (OE , gE ). Proof. With analytic changes of chart we can write near each of the centers q = Q 2 , then the metric ĝE can be explicitly computed showing that it is non degenerate on the singularity and thus can be easily extended. 2.2. BIFURCATION DIAGRAMS 23 In case RE is not empty the trajectory must be regularized on its boundary, for this purpose the regularization technique by Seifert [Sei48] can be applied to obtain geodesic segments of finite length that reaches the boundary of RE perpendicularly and that can be continued by reflection on the boundary. The rest follows as in [KK92, Theorem 3.1]. □ Remark 2.6. If Z1 and Z2 are both negative, the flow ΦEt is already complete since the particle with finite energies cannot reach the centers. On the other hand, if there is only one index i+ ∈ {1, 2} such that Zi > 0, + the particle is still able to reach the center at si : if we are in one of the case not + described by the previous theorem, the motion must be regularized. One of the many ways in which this can be done to extend it by reflecting the particle backward when it collides with the nucleus si . In this way the continuity with respect to the initial + conditions is preserved. To implement rigorously the regularization, it is enough to parametrize the state of the particle with its energy and its incoming and outgoing directions. The completion of the phase space is then obtained adjoining to it the cylinder R × S 1 , see for example [Kna11, Theorem 11.23] or [KK92, Proposition 2.3]. 2.2. Bifurcation diagrams The topological structure of a Hamiltonian motion can be described by means of the constants of motion of the system. Taken together, the constants of motion define a vector valued function on the phase space of the Hamiltonian. We can study the structure of the preimages of this function (its levelsets) and how the constants of motion relate together. In the simplest and rather untypical case the levelsets are diffeomorphic manifolds. The general picture is local and can be described through the following local structures [Kna11, Section 7.3]. Given two manifold M and N let f ∈ C ∞ (M , N ). We say that f is locally trivial at y0 ∈ N if there exists a neighborhood V ⊆ N of y0 such that f −1 (y) is a smooth submanifold of M for all y ∈ V and there there is a map g ∈ C ∞ ( f −1 (V ), f −1 (y0 )) such that f × g : f −1 (V ) → V × f −1 (y0 ) is a diffeomorphism. Notice that if f is locally trivial, the restriction g ↾ f −1 (y) : f −1 (y) → f −1 (y0 ) is a diffeomorphism for every y ∈ V . The bifurcation set of f is the set B( f ) := {y0 ∈ N | f is not locally trivial at y0 }. Remark 2.7. It is not difficult to prove that the critical point of f are points of B( f ) (see [Kna11, Lemma 7.24]), on the other hand the converse is true only in the case f is proper (i.e. it has compact preimages). 2 A simple counterexample is given by f : R → R, f (q) := −e −q . In its case the minimum −1 is the only singular point of f but B( f ) = {0, −1}. 24 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS Define the function on the phase space as follows H ∗ 3 : T (R \ {±a}) → R3 . H F := ξ pϕ (2.2.1) In what follows we want to characterize the bifurcation set B(F ). 2.3. Bifurcations for planar motions We consider here the case l = 0, corresponding to the planar motions on the (q1 , q2 )-plane, define F0 the restriction of F ↾ p −1 (0) . We have already discussed the ϕ regularizability of the problem, for what follows we proceed similarly as [WDR04] but we consider the energy range E ≥ 0. To cover the (q1 , q2 )-plane of the configuration space Q2 := R2 \ {±a} we need two half strips [1, ∞) × [−1, 1] with ϕ = 0 and ϕ = π (i.e. one for each sign of q2 ). Alternatively we can take the cylinder (ξ , η) ∈ R × [−π, π] as the modified configuration space Q 2 : it is a two-sheeted cover with branch points at the foci that could remind the previous section. The two sheets are related by the involution I : (ξ , η) 7→ (−ξ , −η) leaving the cartesian coordinates (q1 , q2 ) unchanged. The symplectic lift of I to the phase space T ∗ Q 2 equals Ib : ( pξ , pη , ξ , η) 7→ (− pξ , − pη , −ξ , −η). Then T ∗ Q2 is obtained from T ∗ Q 2 by factorization with respect to I . Remark 2.8. An analysis of the extrema of Vξ and Vη for l = 0 implies that the image R of (H , Hξ ) in R2 is bounded by the following curves. From K = Hξ ≥ Vξ we have K ≥ K+ (E) with −∞, E >0 Z −(Z + E), E ≤ min − 2+ , 0 K+ (E) := + Z Z+2 , 0 ≥ E > min − 2+ , 0 4E (2.3.1) and from −K = Hη ≥ Vη we have K ≤ K− (E) with K− (E) := Z− − E, E ≤ Z−2 , 4E E> Z− 2 Z− 2 . (2.3.2) 2.3. BIFURCATIONS FOR PLANAR MOTIONS 25 The main objects of our analysis are transformed by the hamiltonian symplectomorphism defined in Theorem 2.3 as follows F (ξ , η) 7→ F̂ (x, y) := x 2 − y 2 , Vξ (ξ ) 7→ V x (x) := −Z+ x − E x 2 + Vη (η) 7→ Vy (y) := Z− y + E y 2 + Hξ ( pξ , ξ ) 7→ H x ( p x , x) := Hη ( pη , η) 7→ Hy ( py , y) := (x 2 −1) p x2 2 (1−y 2 ) py2 2 l2 , 2(x 2 −1) l2 , 2(1−y 2 ) (2.3.3) + V x (x), + Vy (y). For the rest of the analysis we proceed with these transformed equations (2.3.3) keeping always in mind their relation with the (ξ , η) variables. Theorem 2.9. Let (Z1 , Z2 ) ∈ R∗ × R∗ , then B (F0 ) ↾E≥0 = (E, K) ∈ L | E ≥ 0 and K+ (E) ≤ K ≤ K− (E) . Here L := L0 ∪ L−1 ∪ L−2 ∪ L−3 ∪ L+2 ∪ L+3 with L0 := {E = 0}, L−1 := {K = Z− − E}, L+2 := {K = −Z+ − E}, L−2 := {K = −Z− − E}, L+3 := {4EK = Z+2 }, L−3 := {4EK = Z−2 }, (2.3.4) and K+ and K− are defined by (2.3.1) and (2.3.2). Proof. The fact that K+ (E) ≤ K ≤ K− (E) is a consequence of Remark 2.8. {E = 0} corresponds to threshold between compact and non compact energy surfaces, therefore it is clear that it belongs to the bifurcation set. By definition, the critical points of F0 are in B (F0 ) ↾E≥0 . To compute them we can take advantage of the simple form of the levelset equation in the (x, y) coordinates. To cover the plane we need to consider the two half strips with ϕ = 0 and ϕ = π (see Remark 1.5). We start assuming ϕ = 0. We can rewrite F0 ( p x , py , x, y) = in the form f1 ( py , y) f2 ( p x , x) H ( p x , py , x, y) H x ( p x , x) = E K (1−y 2 ) py2 2 0 + Z− y + E y K+ 2 := = (x 2 −1) p x2 0 K− + Z+ x + E x 2 2 and use of this last representation to compute the critical points. We look for values of ( p x , py , x, y) such that −(x 2 − 1) p x 0 0 (1 − y 2 ) py f1 ( py , y) D = −x p 2 + Z + 2E x 0 f2 ( p x , x) + x 2 0 −y py + 2E y + Z− t 26 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS has rank smaller than 2. It is then a simple exercise to check that the critical points are given by the values Æ (1) ( p x , ± Z− ± 2E, x, ±1), (2) ( p x , 0, x, −Z− /2E), Æ (3) (± 2E + Z+ , py , 1, y), (4) (0, py , −Z+ /2E, y). Substituting these values in the equations f1 ( py , y) = 0 and f2 ( p x , x) = 0 one obtains the following curves in the (E, K) plane (1) K ± Z− + E = 0, thus L−1,2 is in B (F0 ), Z2 (2) K − 4E− = 0, thus L−3 is in B (F0 ), (3) K + Z+ + E = 0, thus L+2 is in B (F0 ), (4) K − Z+2 4E = 0, thus L+3 is in B (F0 ). For what concerns the half strip (x, y) with ϕ = π, it is enough to notice that it corresponds to the strip ϕ = 0 with the inversion ( p x , py , x, y) 7→ ( p x , − py , x, −y). Therefore it reduces to the analysis that we already performed. We want to show that for energy parameters (E, K) in a connected component of R \ B(F0 ), the energy levels F0−1 (E, K) are diffeomorphic. We start discussing a special example. Let (Z+ , Z− ) = (0, 2). Let (E0 , K0 ) be in the interior of the region bounded by L−1 and K− (E) (see Figure 2.1, bottom left plot). We will show in Section 2.5 that all the trajectories in configuration space with energy (E0 , K0 ) must cross a segment S0 strictly contained in the segment joining the the two centers (see Figure 2.4, plot 5 and 6 counted from the left). Since (E0 , K0 ) ̸∈ L the crossing must be transversal. Therefore by the linearizability of the vector field we can define a Poincaré section S0 , such that every trajectory is uniquely identified by its crossing point (see [Kna11, Satz 3.46 and Definition 7.16]). Let (E1 , K1 ) be another point in the interior of the region containing (E0 , K0 ). As before there is a segment S1 strictly contained in the segment joining the two centers that is crossed transversally by all the trajectories (with energy (E1 , K1 )). Given (E1 , K1 ) we can define the Poincaré section S1 and every point on the levelset is identified by its crossing point and its time, thus the levelset is diffeomorphic to R × S1 . Clearly, S0 and S1 are diffeomorphic and thus the levelsets of (E0 , K0 ) and (E1 , K1 ) are diffeomorphic. By the generality of (E0 , K0 ) and (E1 , K1 ) all the points in the interior of the region bounded by L−1 and K− (E) have diffeomorphic levelsets. We consider another example, again (Z+ , Z− ) = (0, 2). Let (E0 , K0 ) be in the interior of the region of {E > 0} bounded by L+2 , L−1 (see Figure 2.1, bottom left plot). We continue to refer to Section 2.5 when we show that for such (E, K) all the trajectories in configuration space must cross a line segment L0 strictly contained in the q1 -axis with q1 < −1 (see Figure 1.1 and the first plot from the right in Figure 2.4). 2.3. BIFURCATIONS FOR PLANAR MOTIONS 27 As in the previous example the trajectories must cross L0 transversally and we can reduce the phase space to the Poincaré section L0 . If (E1 , K1 ) is another point in the same region we can reiterate the procedure to find a Poincaré section L1 that is diffeomorphic to L0 . And thus all the points in the region have diffeomorphic levelsets. The argument sketched above can be reproduced in each connected component of R \ B(F0 ) choosing a proper transversal section. How to make the choice will be clear in the next three sections, where we characterize the motion in configuration space for the energy parameters in each region. □ Remark 2.10. Differently from L0 = {E = 0}, the line {K = 0} is in the bifurcation set only in the symmetric case Z− = 0. In this case, in fact, it corresponds to the boundary K− (E) of the Hill’s region. Of course there may be points {K = 0} in the bifurcation set for Z− ̸= 0, but these are just the points in which the curves in L cross transversally {K = 0} (see Figures 2.5 and 2.6). Remark 2.11. The characterization of the bifurcation set given in Theorem 2.9 is redundant. Namely some of the curves L∗∗ restricted to the values of (K, E) in the Hill’s region could be empty for some values of Z1 and Z2 . For example, being K ≤ K− (E), we can immediately see that the curve L+3 will be in the bifurcation diagram for positive E only when Z+ < 0 and |Z+ | < Z− . In what follows we will describe more precisely the structure of the bifurcation sets and of the trajectories in configuration space in relation to the values assumed by Z+ and Z− . The momenta ( p x , py ) at given (E, K) are given in general by p x2 py2 = = 2(x 2 − 1)(E x 2 + Z+ x + K) − l 2 (x 2 − 1)2 −2(1 − y 2 )(E y 2 + Z− y + K) − l 2 (1 − y 2 )2 , (2.3.5) . The Hill’s region is identified by the values of E and K that admits non-negative squared momenta, being the denominator always positive we can discuss them and identify the possible motion types in terms of the zeros of the polynomials P l (s ) := 2(s 2 − 1)(E s 2 + Z± s + K) − l 2 (2.3.6) where s ∈ {x, y} and the understanding that we choose “ +” for s = x and “ −” for s = y . The factor (s 2 −1) is introduced to provide the correct signs and for computational convenience. The momenta can be simply obtained via (x 2 − 1) p x = ± Æ P 0 (x) and 28 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS Æ P 0 (y). The roots of P+0 (x) and P−0 (y) are respectively s Z+2 Z+ K x1,2 = ±1, x3,4 = − ± − , 2E 4E 2 E s Z−2 Z− K y1,2 = ±1, y3,4 = − ± − , 2 2E E 4E (1 − y 2 ) py = ± (2.3.7) with the convention that the smaller index correspond to the solution with negative sign. In both variables, the polynomials have two fixed roots at ±1 and two movable roots which depend on the constants of motion. Being x ∈ [1, ∞), we are going to consider only roots in this region. The discriminant of P 0 is proportional to discr(P 0 ) = (Z±2 − 4EK)(E + K − Z± )2 (E + K + Z± )2 . (2.3.8) Double roots appear when discr(P 0 ) vanishes. For each couple (Z+ , Z− ) this gives six curves in the (K, E)-plane, three for the x variable and three for the y . These are the curves L+1 := {K = Z+ − E} and L−1 , L±2,3 defined by (2.3.4). Remark 2.12. The zeroes of the discriminant P 0 (2.3.8) corresponds to the double roots of (E s 2 + Z± s + K) and the points in which these roots reach the fixed roots ±1, that is the q1 -line. The positivity of P 0 and the positions of its roots, as we will see, characterizes the trajectories in configuration space. The curve L+1 appearing in the discriminant depends from the fact that we considered x ∈ R. As such it will have no correspondence in the bifurcation set or in the description of the possible motions. As a first step we consider the cases in which Z− = 0 or Z+ = 0. In these cases the (K, E)-plane is divided by the curves L±1,2,3 into different regions. We will label these regions using roman numbers with a subscript chosen between >, < and 0 indicating if Z+ > 0, Z+ < 0 or Z+ = 0 respectively. In Figure 2.1 are shown representative bifurcation diagrams for these three cases with the corresponding enumeration of the regions. 2.4. Motion for Z− = 0 The case Z− = 0 corresponds to two attracting (or repelling) centers with the same charges. We have the following corollary of Theorem 2.9. Corollary 2.13. Let Z− = 0 and Z+ ∈ R∗ . With the notation of (2.3.4) we have ¦ © B(F0 )↾E≥0 = (E, K) ∈ L0 ∪ L−1 ∪ L+2 | E ≥ 0 and K+ (E) ≤ K ≤ 0 . Proof. In (1.1.1) we assumed Z1 , Z2 ̸= 0, therefore Z− = 0 implies Z+ ̸= 0. K cannot be positive because K− (E) = 0 for E ≥ 0. By L−1 = L−2 and L−3 = L0 , it is redundant to add both in the definition of the bifurcation diagram. The fact that L+3 is not in the bifurcation set follows from Remark 2.11. Then the claim follows directly from Theorem 2.9. □ 2.4. MOTION FOR Z− = 0 L−1,2 L−1,2 E 29 L+2 E I I I< L+2 I I I> I I< I I> I> K (Z+ , Z− ) = (2, 0) L+2 E (Z+ , Z− ) = (2, 0) (Z+ , Z− ) = (−2, 0) I V0 I I I0 I I0 I0 K (Z+ , Z− ) = (−2, 0) L−1 L−2 I< I V0∗ I I I0∗ (Z+ , Z− ) = (0, 2) K (Z+ , Z− ) = (0, 2) Figure 2.1. Examples of bifurcation diagrams for the considered planar cases, the shaded regions identify the complement of the Hill’s region, the red dashed curves are K+ and K− . The green line L+2 parametrized by K0 (E) = −(Z+ +E) corresponds to the closed orbit wandering between the two centers and having coordinate x = 1. The box on the right shows the shape of the potential V on the q1 -axis in the different cases. Consider the bifurcation diagram for (Z+ , Z− ) = (±2, 0) appearing in Figure 2.1. It is possible to describe the qualitative structure of the motion in configuration space for energy parameters in the different regions by studying the dynamic of the roots (2.3.7) with respect to a walk in the bifurcation diagram. By this we mean fixing a value of E big enough and varying K to move through the regular regions I>,< , I I>,< , I I I>,< and to cross the bifurcation lines. A qualitative representation of how the motion changes with respect to the energy parameters is shown in Figure 2.2 for Z+ = 2 and in Figure 2.3 for Z+ = −2. This can be schematically explained through the behavior of the roots (2.3.7) as follows. 30 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS HK, EL = H-6, 3L HK, EL = H-5, 3L HK, EL = H-4, 3L HK, EL = H-3, 3L HK, EL = H-1, 3L Figure 2.2. Example of possible trajectories in the case Z+ = 2, Z− = 0 for E = 3 and growing values of K (from left to right) chosen in the different regions of the bifurcation diagram. • Roots of the polynomial P 0 (y). – For energies in the regions I>,< and I I> of the bifurcation diagrams, the polynomial P 0 (y) is positive for every value of y ∈ [−1, 1] and |y3,4 | > 1. Thus for energy parameters in these regions, the particle is allowed travel in configuration space everywhere in a region around the centers. – Line L−1 characterizes the values (E, K) such that the two group of roots of P 0 (y) merge: y3 = y1 = −1 and y4 = y2 = 1. – In the regions I I< and I I I>,< , |y3,4 | < 1 and the P 0 (y) is not negative only if y ∈ [y3 , y4 ]. This means that the motion in configuration space can cross the axis between the two centers only passing through the segment joining them centers. • Roots of the polynomial P 0 (x). – Notice that x1 and x3 are always smaller than 1, thus they won’t enter the discussion. – In the regions I>,< and I I< , the root x4 > 1 and P 0 (x) is positive only if x ∈ [x4 , ∞). Therefore in configuration space, the particle cannot reach the line joining the two centers. – For (E,V ) ∈ L+2 , we have the collision of the solutions x4 = x2 = 1 and in configuration space the particle can reach the line between the centers. – On the right of L+2 , in the regions I I> and I I I>/< , the root x4 < 1 and P 0 (x) is non-negative for x ∈ [1, ∞). In other words the particle can cross the line in configuration space connecting the centers. We can now understand the peculiarity of the lines in the bifurcation set. • For values of the parameters on the singular line L−1 we can identify two special trajectories in configuration space. In these, the particle lies in the positive (negative) q1 -axis with |q1 | > 1, possibly bouncing against the singularity and being reflected back. • For (E,V ) on L+2 we can find the unique periodic orbit of the regularized classical two-centers problem, see [Kna02]. It is the hyperbolic trajectory of a particle bouncing between the centers. Counting from left to right, 2.5. MOTION FOR Z+ = 0 31 the second plot of Figure 2.2 and the fourth plot of Figure 2.3 show the trajectory of a particle on the stable manifold of this special orbit. • For K = 0 (E > 0) only one trajectory is possible: the vertical trajectory moving on the line y = 0. HK, EL = H-6, 3L HK, EL = H-3, 3L HK, EL = H-2, 3L HK, EL = H-1, 3L HK, EL = H-0.5, 3L Figure 2.3. Example of possible trajectories in the case Z+ = −2, Z− = 0 for E = 3 and growing values of K (from left to right) chosen in the different regions of the bifurcation diagram. The red line corresponds to the energy level of the plotted trajectory. Remark 2.14. Notice that the ordering of the singular curves reflects the main difference between the cases Z+ > 0 and Z+ < 0. In the first case (corresponding to the attracting potential) the particle is able to travel arbitrarily near to the centers. In the case Z+ < 0 the centers ±a have a positive distance from the Hill’s region. 2.5. Motion for Z+ = 0 Corollary 2.15. Let Z+ = 0 and Z− > 0. With the notation of (2.3.4) we have n B(F0 )↾E≥0 = (E, K) ∈ L0 ∪ L−1 ∪ L−2 ∪ L−3 ∪ L+2 | o E ≥ 0 and K+ (E) ≤ K ≤ K− (E) . Proof. In (1.1.1) we assumed Z1 , Z2 ̸= 0, therefore Z+ = 0 implies Z− ̸= 0. □ We have L+3 = L0 . The claim follows directly from Theorem 2.9. As in the previous section we give a qualitative explanation of the possible motions in configuration space through the behavior of the roots (2.3.7). A visual support is provided by Figure 2.4. • Roots of the polynomial P 0 (y). – For (E,V ) in I0 , the polynomial P 0 (y) is not negative for any y ∈ [−1, 1]. Thus in configuration space the particle is free to move around the centers. – For energy parameters on L−2 , two roots collide: y4 = y2 = 1. – The motion in configuration space for (E,V ) in I I0 and I I I0 is restricted to y ∈ [−1, y4 ]. That is, the particle is free to travel around the attracting center but is bounded away from the repelling one. – For energies on L−1 , the other two roots collide: y3 = y1 = −1 and for (E,V ) ∈ I V− the only allowed y are restricted in y ∈ [y3 , y4 ]: 32 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS in configuration space the particle cannot anymore travel around the centers. – On the line K = 0, y4 = 0 and for bigger values of K (i.e. in the regions I I I0∗ and I V0∗ ) y4 becomes negative. The particle in configuration space is no more able to flow around the repelling center. • For the roots of P 0 (x) the discussion is similar as before. – The roots x1,3 are negative. We consider only the roots x2,4 . – For energy parameters in I0 and I I0 the root x4 > 1 and the polynomial P 0 (x) is positive for x ∈ [x4 , ∞). Therefore in configuration space the particle cannot reach the line between the centers. – For energy parameters on the right of L+2 the motion becomes possible for x ∈ [1, ∞). I.e. the particle can reach the line between the centers. HK, EL = H-6, 3L HK, EL = H-4, 3L HK, EL = H-3, 3L HK, EL = H-2, 3L HK, EL = H0, 3L HK, EL = H0.2, 3L HK, EL = H0.5, 0.5L Figure 2.4. Example of possible motions in the case Z+ = 0, Z− = 2 for values of E and K in different regions of the bifurcation diagram. The red line is the energy level of the trajectory in the plot. 2.6. The general case We first describe the bifurcation set for the fully repelling (or attracting) configuration sign(Z1 ) = sign(Z2 ). The picture is similar to the one with Z− = 0 (see Section 2.4) with the only difference that some positive values of K are allowed (see Figure 2.5). Corollary 2.16. Let |Z+ | > Z− , Z− ∈ R0+ . With the notation of (2.3.4) we have n B(F0 )↾E≥0 = (E, K) ∈ L0 ∪ L−1 ∪ L−2 ∪ L−3 ∪ L+2 | E ≥ 0, o K+ (E) ≤ K ≤ K− (E) . Proof. By Remark 2.11, L+3 is not in the bifurcation set. The corollary follows □ immediately from Theorem 2.9. The structure of the trajectories and the qualitative behavior of the motion in configuration space for this case is analogous to the one presented in Section 2.4, therefore we will not discuss it. The case |Z+ | < Z− looks more complicated and particularly interesting. Corollary 2.17. Let |Z+ | < Z− , Z− ∈ R0+ . If Z+ > 0 we have n B(F0 )↾E≥0 = (E, K) ∈ L0 ∪ L−1 ∪ L−2 ∪ L−3 ∪ L+2 | o E ≥ 0 and K+ (E) ≤ K ≤ K− (E) , 2.6. THE GENERAL CASE (Z+ , Z− ) = (4, 2) 33 (Z+ , Z− ) = (−4, 2) Figure 2.5. Bifurcation diagrams for the fully attracting (left) and the fully repelling (right) case respectively. The shaded regions identify the complement of the Hill’s region. while if Z+ < 0 we have n B(F0 )↾E≥0 = (E, K) ∈ L0 ∪ L−1 ∪ L−2 ∪ L−3 ∪ L+2 ∪ L+3 | o E ≥ 0 and K+ (E) ≤ K ≤ K− (E) . Proof. By Remark 2.11, L+3 is not in the bifurcation set. The corollary follows □ immediately from Theorem 2.9. In this case the qualitative behavior of the motion in configuration space is analogous to the one presented in Section 2.5, but for energy parameters in the region I<a . This region contains the set of energy parameters included in the region n bounded above by L+3− := (E, K) ∈ L+3 E < and below by E ≥ 0 (see Figure 2.6). L−1 L+2 I V> L+3 I I I> I> (Z+ , Z− ) = (2, 4) o L+2 I I> L−2 |Z+ | 2 L−2 , on the sides by L+2 and L+1 L−1 L+3 I V> I I I< I<a I<b I< I I< (Z+ , Z− ) = (−2, 4) Figure 2.6. Bifurcation diagrams and labeled regions for |Z+ | < Z− . The shaded regions identify the complement of the Hill’s region. For (E,V ) on I<a a new phenomenon appears: both the movable roots of P 0 (x) are bigger than one and the polynomial is positive in the union of the two disjoint 34 2. THE CLASSICAL PROBLEM OF TWO COULOMB CENTERS intervals [1, x3 ] and [x4 , 1]. On the configuration space they gives rise to an escaping trajectory similar to the previous ones and to a family of bounded trajectories near the attracting center (see Figure 2.7). HK, EL = H0.5, 1.5L HK, EL = H1, 1L HK, EL = H2, 0.5L HK, EL = H2, 0.3L HK, EL = H2, 0.25L Figure 2.7. Bounded motions for Z+ < 0, |Z+ | < Z− and E ≥ 0. From left to right: trajectory for energy parameters on the boundary L+2 of I<a , trajectory for energies on the tangency point between L+2 and L+3 , trajectory for energies on the boundary L+3 of I<a Then follow two trajectories for energies inside I<a and the trajectory on L+3 . The red line is the energy level of the trajectory in the plot, the dotted line is the 0-energy level. HK, EL = H2, 0.5L CHAPTER 3 The two-centers system on L2 (R3 ) As a first step we want to analyze from an operator theoretical point of view the separation of variables obtained in the previous section and its consequences on the nature and the properties of the resulting equations. 3.1. First separation from an operator theoretical point of view We transform by the diffeomorphism G from (1.2.1) into prolate elliptic coordinates (ξ , η, ϕ) with the unitary operator denoted by G : L2 (R3 , d q) → L2 (M , d χ d ϕ). (3.1.1) We index the transformed objects by G . Let us write χ = (ξ , η) for the first two prolate elliptic variables and d χe := sinh ξ sin η d ξ d η, d χ = F (ξ , η)d χe, (3.1.2) where F (ξ , η) = sinh2 (ξ ) + sin2 (η) is the map given by (1.2.2). The weight in the measure d χ is nothing else than the Jacobian of the change of coordinate defined by (1.2.1). The weight in the measure d χe equals the Jacobian for the planar case. So d χ is a measure on the domain R+ × (0, π), absolutely continuous with respect to the Lebesgue measure λ2 . Likewise, d χe is a measure on M (defined in Section 1.2) absolutely continuous with respect to the Lebesgue measure λ3 Proposition 3.1. The operator H = −h 2 ∆ + V on L2 (R3 ) (defined in (1.1.1)) is unitarily equivalent to the operator in L2 (M , d χ d ϕ) given by HG := −h 2 ∆G + VG (ξ , η) where ∆G = 1 ( 1 F (ξ , η) sinh(ξ ) + 1 sin(η) ∂ξ sinh(ξ ) ∂ξ + ∂η sin(η) ∂η + The potential is given by VG (ξ , η) = − Z+ cosh(ξ ) − Z− cos(η) F (ξ , η) 35 , 1 sin2 (η) + 1 sinh2 (ξ ) ! ) ∂ϕ2 . 36 3. THE TWO-CENTERS SYSTEM ON L2 (R3 ) and F is defined by (1.2.2). HG is essentially self-adjoint on n G C0∞ (R3 ) = f (ξ , η, ϕ) ∈ C0∞ M | π-periodic in η, 2π-periodic in ϕ and ∂ξ f |ξ =0 = 0 o and self-adjoint on G (H 2 (R3 )). Proof. It is well known (see for example [RS75, Theorem X.15, p. 165] or [HS96, Theorem 13.7, p. 136]) that H is essentially self-adjoint on C0∞ (R3 ) (and self-adjoint on H 2 (R3 )), see Remark 3.2 below. It is obvious that the unitarily transformed operator HG := G H G −1 remains essentially self-adjoint on G C0∞ (R3 ) , then the form of the operator HG is given by the computations in the proof of Theorem 1.7. □ Remark 3.2. That H is self-adjoint on the Sobolev space H 2 (R3 ), is a straightforward application of the Kato-Rellich Theorem, see Appendix C. On one hand there is the self-adjointness of the Laplacian on H 2 (R3 ), on the other hand there is the ∆-boundedness with relative bound 0 of the potentials V ∈ L2 (R3 ) + L∞ (R3 ) (see [RS75, Theorem X.15]). In this last point is hidden the relevant fact of this proof: if we write V = V2 + V∞ and we take ψ ∈ D(V ) we have ∥V ψ∥2 ≤ ∥V2 ∥2 · ∥ψ∥∞ + ∥V∞ ∥∞ · ∥ψ∥2 , but in principle we have no information on ∥ψ∥∞ . Both D(V ) and D(−∆) contains C0∞ (R3 ), then it follows that we can use the characterization of functions in the domain of the laplacian (see [RS75, Theorem IX.28]), namely ψ is a bounded continuous function and for any a > 0, there is a b , independent of ψ, so that ∥ψ∥∞ ≤ a∥∆ψ∥2 + b ∥ψ∥2 and we have the ∆-boundedness in L2 (R3 ). Notice that the Coulombic potentials are in this class only if the dimension of the space is strictly bigger than 2. In fact for operators in L2 (R) or L2 (R2 ) this is no more true. In particular in dimension 2 the self-adjointness can still be proved using KLMN Theorem [RS75, Theorem X.17] and the infinitesimally form boundedness of V w.r.t. ∆ [Agm82, Theorem 3.2]. In this way the operator is well defined and has form domain H 1 (R2 ). In one dimension even this weaker construction fails and it is necessary to remove the singularities from the space and fix proper boundary conditions on the domain. Define M0 := R0+ ×[0, π]. Let us denote D the space of finite linear combination of products f (χ )w(ϕ) where ¦ © b ∞ (M ) := f (ξ , η) ∈ C ∞ (M ) | π-periodic in η, ∂ f | f ∈C = 0 0 0 ξ ξ =0 0 0 ⊂ L2 (M0 , d χ ) (3.1.3) and w ∈ C ∞ (S 1 ). Due to the factorizability of L2 spaces (see [RS80, Theorem II.10, p. 52]) and the decomposition in prolate elliptic coordinates L2 (M , d χ d ϕ) ∼ = L2 (M0 , d χ ) ⊗ L2 (S 1 , d ϕ), 3.2. A REMARK ON THE SECOND SEPARATION 37 G −1 D is a dense set in L2 (R3 ). Furthermore for functions ψ(χ , ϕ) = f (χ )w(ϕ), the operator H takes the form H f (χ )w(ϕ) = −h 2 Aχ f (χ ) w(ϕ)+ with Aχ := 1 ( 1 F (ξ , η) sinh(ξ ) +h −2 h 2 f (χ ) sinh2 (ξ ) sin2 (η) ∂ξ sinh(ξ ) ∂ξ + Z+ cosh(ξ ) − h −2 1 sin(η) B w(ϕ) (3.1.4) ∂η sin(η) ∂η + ) Z− cos(η) acting on L2 (M0 , d χ ) and B := Bϕ := −∂ϕ2 the Laplace-Beltrami operator acting on L2 (S 1 , d ϕ). The operator B is essentially self-adjoint on C ∞ (S 1 ) and admits an orthonormal basis of eigenfunctions, namely Bωλ (ϕ) = λ2 ωλ (ϕ), λ ∈ Z, p where ωλ (ϕ) = e i λϕ / 2π. The family of ωλ defines an orthonormal basis of L2 (S 1 , d ϕ). Call Ωλ the subspace spanned by ωλ and define L2λ := L2 (M0 , d χ ) ⊗ Ωλ to have L2 (R3 ) ∼ = ⊕ λ∈Z L2λ . Then, if 1lλ is the identity operator on Ωλ , the restriction of H to Dλ = D ∩ L2λ is given by H Dλ = Hλ ⊗ 1lλ , where Hλ := −h 2 F (ξ , η) Aχ − λ2 sinh2 (ξ ) − λ2 sin2 (η) (3.1.5) acts on L2 (M0 , d χ ). This leads to the proposition that follows. Proposition 3.3. The operator H defined in (1.1.1) is unitarily equivalent ⊕ ⊕ to the operator λ∈Z Hλ ⊗ 1lλ acting on λ∈Z L2λ . 3.2. A remark on the second separation The problem is now transferred to the study of the self-adjoint extension Hλ of each operator in the new family defined by Proposition 3.3. b ∞ (M ) we have Notice that for φ, ψ ∈ C 0 0 〈ψ, Hλ φ〉L2 (M 0 ,d χ ) = 〈ψ, Kλ φ〉L2 (M e) 0 ,d χ 3. THE TWO-CENTERS SYSTEM ON L2 (R3 ) 38 where Kλ = F (ξ , η)Hλ , i.e. Kλ = −h 2 Aχ − λ2 sinh2 (ξ ) − λ2 sin2 (η) . (3.2.1) In the same way, if we consider the eigenvalue equation for Hλ , we have that b ∞ (M ) for ψ, ϕ ∈ C 0 0 〈ψ, (Hλ − E)ϕ〉L2 (M 0 ,d χ ) = 〈ψ, (Kλ − E F (ξ , η))ϕ〉L2 (M e) 0 ,d χ , and consequently (Hλ − E)ϕ(ξ , η) = 0 ⇐⇒ (Kλ − E F (ξ , η))ϕ(ξ , η) = 0. The differential equation (Kλ − F (ξ , η)E)ψ(ξ , η) = 0 can be rewritten as (Kλ − F (ξ , η)E)ψ(ξ , η) = F (ξ , η)(Hλ − E)ψ(ξ , η) = Aξλ,η ψ(ξ , η) = 0 (3.2.2) where λ Aξλ,η (h) := Aξλ,η,h,E := (Aξλ,h,E + Aη,h,E ) (3.2.3) with Aξλ (h) := −h 2 1 sinh(ξ ) ∂ξ sinh(ξ ) ∂ξ − Z+ cosh(ξ ) − E cosh2 (ξ ) + h 2 λ2 sinh2 (ξ ) and Aηλ (h) := −h 2 h 2 λ2 ∂η sin(η) ∂η + Z− cos(η) + E cos2 (η) + 2 . sin(η) sin (η) 1 We will call Aξλ (h) radial operator and Aηλ (h) angular operator, names coming by an obvious comparison with role of the variables ξ and η in the prolate spheroidal coordinate (see Remark 1.5). In view of this fact, it seems natural to move our point of view from the measure space L2 (M0 , d χ ) to L2 (M0 , d χe) = L2 (M0 , sinh ξ sin η d ξ d η) ∼ = L2 (R0 , sinh ξ d ξ ) ⊗ L2 ([0, π], sin η d η). + In the next sections we will study these two last operators acting respectively on L2 (R0+ , sinh(ξ ) d ξ ) and L2 ([0, π], sin(η) d η) to justify properly the weak separation described in this section. 3.3. Self-adjoint extension and spectrum of the angular operator Set I0 := [−1, 1]. If we apply the unitary transform L2 ([0, π], sin η d η) → L2 (I0 , d x), g (η) 7→ g (arccos(x)), (3.3.1) the operator Aηλ (h) transforms to A xλ (h) := L xλ (h) + V (x) (3.3.2) 3.3. SELF-ADJOINT EXTENSION AND SPECTRUM OF THE ANGULAR OPERATOR 39 where h 2 λ2 L xλ (h) := −h 2 ∂ x (1 − x 2 )∂ x + 1 − x2 and V (x) := E x 2 + Z− x. (3.3.2a) The eigenvalue equation corresponding to L xλ (1) L xλ (1)u = k(k + 1)u is the associated Legendre equation. For k ∈ N its eigenfunctions are the associated Legendre function Pkλ (x) := (−1)λ (1 − x 2 )λ/2 ∂ xλ Pk (x), |λ| < k, where the Pk (x) are the Legendre polynomials Pk (x) := 1 k 2 k! ∂ xk (x 2 − 1)k . Another solution, linearly independent from Pkλ , is given by Qkλ (x) = Pkλ (x) ∫ x 0 dt (1 − t )(Pkλ (x))2 2 . If we choose the boundary conditions generated by P0 (x) = 1, the following theorem can be proven Theorem 3.4. The operator L xλ (1), λ ∈ Z, defined via D(L xλ (1)) = f ∈ L2 (I0 ) | f , f ′ ∈ AC (I0 ), L xλ (1) f ∈ L2 (I0 ), lim (1 − x 2 ) f ′ (x) = 0 if λ = 0 x→±1 is self-adjoint. Its spectrum is purely discrete, that is σ(L xλ (1)) = σd (L xλ (1)) = {k(k + 1) | k ∈ N0 , |λ| ≤ k}, and the corresponding eigenfunctions s uk,λ (x) = 2k + 1 (k − λ)! 2 (k + λ)! Pkλ (x), k ∈ N0 , |λ| ≤ k, form an orthonormal basis for L2 (I0 ). Proof. The result is standard, one proof can be found in [Tes09, Theorem 10.6, p. 226]. The definition of the domain follows by the fact that L xλ is LCC at both −1 and 1 for λ = 0 and it is LPC in both the points for λ ̸= 0, as one could guess from the presence of the λ dependent singularity at ±1. □ We need the following Corollary of Theorem 3.4. 3. THE TWO-CENTERS SYSTEM ON L2 (R3 ) 40 Corollary 3.5. The operator L xλ (h), λ ∈ Z, defined via D(L xλ (h)) = f ∈ L2 (I0 ) | f , f ′ ∈ AC (I0 ), L xλ (h) f ∈ L2 (I0 ), lim (1 − x 2 ) f ′ (x) = 0 if λ = 0 x→±1 is self-adjoint. Its spectrum is purely discrete, that is σ(L xλ (h)) = σd (L xλ (h)) = h 2 σd (L xλ (1)), and the corresponding eigenfunctions h uk,λ (x) with k ∈ N, k(k + 1)h 2 ∈ σ(L xλ (h)), form an orthonormal basis for L2 (I0 ). Theorem 3.6. The operator A xλ (h) with domain D(A xλ (h)) = D(L xλ (h)) is self-adjoint and its essential spectrum is empty. Proof. We know from Theorem A.5 that the resolvent of L xλ (h) is a HilbertSchmidt operator and thus is compact. In our domain the multiplication operator V is bounded by |V (x)| ≤ |E| + |Z− |, thus it is only a bounded perturbation of the operator L xλ (h). It follows directly that V · (L xλ (h) − i )−1 is compact (for more details see [RS80, Theorem VI.12, p.200]) and thus V is relatively compact with respect to L xλ (h). Therefore by Theorem C.2 follows that A xλ (h) is self-adjoint on D(L xλ )(h) and σe s s (L xλ (h)) = σe s s (A xλ (h)) = ;. □ Theorem 3.7 (min-max principle, operator form). Let H be a self-adjoint operator that is bounded from below, i.e., H ≥ c I for some c . Define µn (H ) := sup φ1 ,...,φn−1 UH (φ1 , . . . , φn−1 ) where UH (φ1 , . . . , φ m ) = inf ψ∈D(H ): ∥ψ∥=1 ψ∈[φ1 ,...,φ m ]⊥ 〈ψ, H ψ〉 and [φ1 , . . . , φ m ]⊥ is a shorthand for {ψ|〈ψ, φi 〉 = 0, i = 1, . . . , m}. Note that the φi are not necessarily independent. Then, for each fixed n , either: (a) there are n eigenvalues (counting degenerate eigenvalues a number of times equal to their multiplicity) below the bottom of the essential spectrum, and µn (H ) is the n th eigenvalue counting multiplicity; or (b) µn is the bottom of the essential spectrum, i.e. µn = inf{λ|λ ∈ σe s s (H )} and in that case µn = µn+1 = µn+2 = · · · and there are at most n − 1 eigenvalues (counting multiplicity) below µn . 3.4. ANALYTIC EXTENSIONS OF THE EIGENVALUES OF Aηλ 41 Proof. The proof of the min-max principle and its consequences are in [RS78, Section XIII.1 p. 75]. □ Given the fact that all the operators are defined on the same domain, one has UL λ (h) + m(Z− , E) ≤ UA λ (h) ≤ UL λ (h) + M (Z− , E), x x (3.3.3) x where m := m(Z− , E) := min V (x) and x∈[−1,1] M := M (Z− , E) := max V (x) (3.3.4) x∈[−1,1] are both finite. As a consequence µn (L xλ )(h) + h −2 m ≤ µn (A xλ (h)) ≤ µn (L xλ (h)) + h −2 M , (3.3.5) thus the spectrum of A xλ (h) is purely discrete and the eigenvalues accumulate at ∞. Moreover Theorem A.5 tells us that the eigenvalues are simple and that the eigenfunctions define an orthonormal basis for L2 (I0 , d x). Remark 3.8. From Theorem 3.4 and its corollary we have in addition that h 2 nλ (nλ + 1) + m ≤ µn (A xλ (h)) ≤ h 2 nλ (nλ + 1) + M where nλ := n + |λ| and n ∈ N, and thus for each fixed value of the small parameter h , if n and/or λ are big enough, we can have some control on the eigenvalues position. Corollary 3.9. Aηλ (h) is self-adjoint on the transformed domain, its spectrum purely discrete and made of simple eigenvalues accumulating at ∞. Proof. It follows as a direct consequence of the unitary equivalence of the operators A xλ (h) and Aηλ (h). □ 3.4. Analytic extensions of the eigenvalues of Aηλ We continue the study of the previously defined operator A xλ (h). All its properties are preserved by the unitary change of variable defined in (3.3.1). The fact that A xλ (h) is closed and x 2 is infinitesimally A xλ (h)-bounded guar- antees that the β-dependent family of operators on D(L xλ (h)) A xλ (β) := A xλ (h, β) := L xλ (h) + V (x) + βx 2 defines an analytic type-(A) family of self-adjoint operators in the sense of Appendix C. In particular it is an entire family because of the L xλ (h)-infinitesimally boundedness of the perturbation. Notice that by definition (3.3.2a) of V , β essentially perturbs the parameter E . In our case we have only non-degenerate, isolated eigenvalues, thus it follows el extends analytically as function of β in from Theorem C.7 that every eigenvalue µ the circle of radius el , ε) = r (0, ∥V ∥, µ ε ∥V ∥ , 3. THE TWO-CENTERS SYSTEM ON L2 (R3 ) 42 where the ε is given by the distance with the nearest eigenvalue, thus it is O(l 2 ) in view of Corollary 3.5. Then, by unitarity the eigenvalues µλl (h) of A λ e (h) extends analytically as η,E,Z− function of E in a neighborhood of the real axis of radius O(l 2 ). By Theorem C.4 the corresponding eigenfunctions extend accordingly. 3.5. Second separation from an operator theoretical point of view We can use the results of Section 3.3 to partially justify the second separation discussed in Section 3.2. Let us denote Dχe the subspace of finite linear combination of products f (ξ )g (η) where f ∈ C0∞ (R0+ ) ⊂ L2 (R0+ , sinh(ξ )d ξ ) with f ′ (0) = 0 and g ∈ C ∞ ([0, π]) ⊂ L2 ([0, π], sin(η)d η) is π-periodic. Due to the isomorphism (see [RS80, Theorem II.10, p. 52]) L2 (M0 , d χe) ∼ = L2 (R0+ , sin(ξ )d ξ ) ⊗ L2 ([0, π], sin(η)d η), b ∞ (M ) in L2 (M , d χe) (see (3.1.3)), we have that D is a dense and the density of C 0 0 χe 0 set in L2 (M0 , d χe). Then for functions ψ(ξ , η) = f (ξ )g (η), the operator Aξλ,η takes the form Aξλ,η f (ξ ) g (η) = Aξλ (h) f (ξ ) g (η) + f (ξ ) Aηλ (h) g (η) . Define µλl := µλl (h, E, Z− ) (3.5.1) the l -th eigenvalue of Aηλ (h) and φ lλ := φ lλ (h, E, Z− ) (3.5.2) the relative eigenfunction. Call Φλl the subspace spanned by φ lλ and define L2l ,λ := L2 R0+ , sinh(ξ )d ξ ⊗ Φλl to have L2 (M0 , d χe) = ⊕ l ∈N L2l ,λ . Then, if 1l l ,λ is the identity operator on Φλl , the restriction of Aλ to D lλ := Dχe ∩L2l ,λ is given by Aξλ,η λ D l = A lλ ⊗ 1l l ,λ , where A lλ := Aξλ + µλl . (3.5.3) This leads to the following proposition. Proposition 3.10. For each integer value of λ, the operator Aξλ,η defined in ⊕ ⊕ (3.2.3) is unitarily equivalent to the operator in l ∈N L2l ,λ given by l ∈N A lλ ⊗1l l ,λ . 3.6. SELF-ADJOINT EXTENSIONS OF THE RADIAL OPERATORS A lλ 43 3.6. Self-adjoint extensions of the radial operators A lλ Let us start analyzing the Weyl criterion at ∞. Set A lλ = Lξ + V lλ (ξ ) (3.6.1) where Lξ := −h 2 1 sinh(ξ ) ∂ξ (sinh(ξ )∂ξ ) and V lλ (ξ ) := V λ (ξ ) := h 2 λ2 2 sinh (ξ ) − E cosh2 (ξ ) − Z+ cosh(ξ ) + µλl (E). We generalize a completeness criterion from [RS75, Theorem X.8]. Theorem 3.11. Consider the differential equation associated to the SturmLiouville operator H x := L x + V (x) in L2 ((0, ∞), p(x)d x) where L x := − 1 p(x) ∂ x ( p(x)∂ x ) and lim p(x) ̸= 0. x→∞ Let V (x) be a continuous real-valued function in (0, ∞) and suppose there exists a positive differentiable function M : R+ → R+ so that (1) V (x) ≥ −M (x) ∫∞ (2) 1 (M (x))−1/2 p(x)d x = ∞ (3) M ′ (x)/(M (x))3/2 is bounded near ∞. Then H x is in LPC at ∞. Proof. We will show that the two solutions of H x ϕ = 0 cannot be both in L2 ( p(x)d x) around ∞. Consider 0 < c1 < c < ∞ and let u be a real solution of the previous equation in L2 ( p(x)d x) near ∞. Then we have ∫∞ ∫c 2 −K1 := − u (x) p(x)d x ≤ − u 2 (x) p(x)d x ∫ ≤ ∫ = c1 c c1 c c1 V (x) M (x) u 2 (x) p(x)d x ∂ x ( p(x)∂ x u(x)) c1 M (x) u(x)d x. That, via integration by parts, gives − u ′ (x)u(x) M (x) c ∫ c (u ′ (x))2 p(x) + p(x)d x c1 c1 M (x) ∫c ′ u (x)u(x)M ′ (x) − p(x)d x ≤ K1 . M 2 (x) c1 (3.6.2) 3. THE TWO-CENTERS SYSTEM ON L2 (R3 ) 44 On the other hand, using 3. and Hölder inequality, we can find K2 such that ∫ − c u ′ (x)u(x)M ′ (x) M 2 (x) ∫c c1 ≤ K2 Suppose that ∫ ∞ (u ′ (x))2 c1 M (x) c1 p(x)d x ≤ (u ′ (x))2 M (x) !1/2 ∫ c p(x)d x !1/2 (3.6.3) (u(x))2 p(x)d x . c1 p(x)d x = ∞, then by the last two inequalities, the positivity of p(x) for x > 0 and the hypothesis on u , u ′ (x)u(x) must be positive near ∞ but this imply that u and its derivative have always the same sign, that is impossible ∫ ∞ (u ′ (x))2 p(x)d x < ∞. Suppose now that ϕ and ψ are two independent solutions of H x ψ = 0 and that both are in L2 ( p(x)d x) near ∞. Moreover, suppose that they are normalized so that ψ(x)ϕ′ (x) − ψ′ (x)ψ(x) = 1. Then, 1/2 1 ϕ(x)ψ′ (x) ϕ′ (x)ψ(x) − = M (x) (M (x))1/2 (M (x))1/2 since u is in L2 ( p(x)d x) near ∞. As a consequence c1 M (x) would be in L1 ( p(x)d x) near ∞ and this contradicts hypothesis 2. . □ λ Observe that the λ term in A l decays fastly at infinity, this should give the hint that it shouldn’t affect the nature of the point as it happened in the analysis of Aηλ . In fact, this is an easy corollary of the previous theorem: Corollary 3.12. The differential equation associated to A lλ is in the LPC at ∞ for any value of λ. Proof. Clearly V λ (ξ ) ≥ −M (ξ ) with M (ξ ) := CE,Z+ ,h,µ cosh2 (ξ ), and this function satisfy all the hypotheses of the last theorem for p(ξ ) = sinh(ξ ). This shows that for A lλ we have LPC at ∞ for every value of λ and µλl . □ For what concerns the nature of the boundary point ξ = 0, we will see that the λ term in A lλ is central. We proceed in a way similar to the one for Aηλ . If we apply the unitary transform L2 (R0+ , sinh(ξ ) d ξ ) → L2 ([1, ∞), d y), g (ξ ) 7→ g (arccosh(y)), (3.6.4) the operator A lλ transforms to Ayλ = Lyλ + V l (y) where, as before, h 2 λ2 Lyλ := Lyλ (h) := −h 2 ∂y (y 2 − 1)∂y + 2 y −1 and V l (y) := −E y 2 − Z+ y + µλl (E). (3.6.5) 3.6. SELF-ADJOINT EXTENSIONS OF THE RADIAL OPERATORS A lλ 45 Near y = 1, V l (y) is a bounded perturbation of the operator Lyλ (1) in the proper domain. Thus it is enough to analyze the limit circle-limit point problem for Lyλ (1) instead of Aλy (1). Similarly as in Section 3.3, we have that ∂y (1 − y 2 )∂y u + k(k + 1) − ! λ2 1 − y2 u =0 is solved by u(y) := c1 Pkλ (y) + c2 Qkλ (y) for all k ∈ N0 s.t. |λ| ≤ k where Pkλ and Qkλ are the associated Legendre functions analytically extended to C \ (−∞, 1] (see [EMOT53, pp. 120-]). What remains to understand is if they are both in L2 ([1, ∞)) or not. In the case λ = 0, as shown in [Leb72, p.167-], we have that limy→1+ Pk0 (y) = 1, thus the function stays bounded even in the closed interval. As a consequence its L2 norm in every compact interval is finite. On the other hand limy→1+ Qk0 (y) = ∞ so we have to check the order of this divergence. For y > 1 (even for | arg(y − 1)| < π) and ℜk > −1, we can represent Qk0 (y) as follows ∫ Qk0 (y) = hk 0 where 1 p hk (w) = y −1 2 cosh ψ d ψ 2 p −1−2k w +1+ w . p w +1 So, considered that hk is monotonic decreasing, e 0 (y) := Q k ∫ 1 0 (3.6.6) hk y −1 4 eψ d ψ y−1 2 cosh2 ψ ≥ y−1 ψ e 4 and that e 0 (y), is explicitly solvable, has a finite L2 norm near 1 and is such that Qk0 (y) ≤ Q k we have that, in any compact interval of the form [1, c], e 0 ∥ < ∞. ∥Qk0 ∥2 ≤ ∥Q k 2 Thus, for λ = 0 we are in LCC at 1. In the case λ > 0, on the other hand, we use the following representations ([EMOT53, p. 157]): Pkλ (y) = Qkλ (y) = Γ(k + λ + 1) πΓ(k + 1) ∫ π (y + (y 2 − 1) cos t )k e i λt d t , 0 2πi Γ(k + λ + 1) 2k+1 Γ(k + 1) e Æ (y 2 − 1)−λ/2 ∫ π (3.6.7) (y + cos t )λ−k−1 (sin t )2k+1 d t , 0 the first equation being valid for λ ∈ N0 , the second for ℜ(k) > −1, ℜ(k +λ+1) > 0. Taken in account that ∫ lim π y→1+ 0 (y + cos t )λ−k−1 (sin t )2k+1 d t = 2λ+k Γ(λ)Γ(k + 1) Γ(k + λ + 1) , 3. THE TWO-CENTERS SYSTEM ON L2 (R3 ) 46 it is clear that both the integral terms are bounded near 1. It is even evident from the remaining terms that, near 1, Pkλ is in L2 while Qkλ is not! This means that for λ ̸= 0 the operator Lyλ (1) and thus Ayλ (1) is LPC at 1. In [EMOT53, p.161 n.10] is shown that for z ∈ C \ (−∞, 1) and every λ and k one has λ (z 2 − 1)∂ z Pkλ (z) = k z Pkλ (z) − (λ + k)Pk−1 (z). With the representation (3.6.7) for Pkλ and this formula for the derivative, it is natural to consider the boundary conditions generated by P10 in 1. We first consider auxiliary operators Lλy,c on the Hilbert spaces L2 (Ic ) for bounded intervals Ic := [1, c). We are not interested in the conditions on the regular point c : these are treated standardly and well defined, we only need to know that there exist a family of initial conditions that must be imposed on the left boundary point to guarantee the (essential) self-adjointness of the operator. As a reference see [Zet05, p. 183-193]. We have proven the following lemma: λ Lemma 3.13. For 1 < c < ∞ define the operator Ly,c (1), λ ∈ Z, by n D(Lyλ (1)) = f ∈ L2 (Ic ) f , f ′ ∈ AC (Ic ), Lyλ (1) f ∈ L2 (Ic ), o lim (1 − y 2 ) f ′ (y) = 0 if λ = 0, BC ( f , c) , y→1+ where BC ( f , c) stands for any separated Boundary Condition for f in the regular λ point c . Then Ly,c (1) is self-adjoint. We have essentially proved the following theorem. Theorem 3.14. The operator Ayλ with domain n D(Ayλ ) = f ∈ L2 (I∞ ) f , f ′ ∈ AC (I∞ ), Ayλ f ∈ L2 (I∞ ), o lim (1 − y 2 ) f ′ (y) = 0 if λ = 0 , y→1+ where I∞ := [1, ∞), is self-adjoint. And thus A lλ is self-adjoint on the domain, transformed by (3.6.4). Proof. First of all notice that the multiplication by a positive constant will not affect the properties of the operator, thus everything said for Lyλ (1) is equally true for Lyλ (h) = h 2 Lyλ (1). One of the main aspects clarified by Theorem A.5 is that the domain of essential self-adjointness of a Sturm Liouville operator depends on the nature of the extremal point of the interval of definition, in this case I∞ . For what concerns ∞, we have already proven in Corollary 3.12 that we have LPC for any λ, thus we do not need to fix boundary conditions. On the other hand V (y) is a bounded perturbation of Lyλ in [1, c], thus we can combine Kato-Rellich 3.7. FORMAL “PARTIAL WAVE EXPANSION” OF THE GREEN’S FUNCTION 47 Theorem and Theorem 3.13 to fix the boundary conditions in 1 required for D(Ayλ ) to define a self-adjoint operator in [1, c]. From this two observations it follows that Ayλ is essentially self-adjoint on D(Ayλ ) as stated in the theorem. The same can be said on A lλ with its transformed domain as a consequence of the unitarity of the transformation. □ 3.7. Formal “partial wave expansion” of the Green’s function For real E we know that the spectrum of Aηλ consists of an infinite number of simple eigenvalues µλ1 (E) < µλ2 (E) < µλ3 (E) < . . . tending to infinity. The µλk extends to analytic functions of E in some neighborhood λ of the real axis. We denote by φn,E the eigenfunctions λ λ Aηλ (E)φn,E (η) = µλn (E)φn,E (η) , n ∈ N0 , normalized by λ ∥2 ∥φn,E ∫ +π = 0 λ (η)|2 sin η d η = 1 |φn,E λ for E ∈ R+ and then extended analytically. We choose φn,E real for E real. Let f ∈ L2 (M0 , d χ ). Instead of solving (Hλ − E)u = f for E ∈ C \ σ(Hλ ), we look at the solution of (Kλ − E F (ξ , η))u(ξ , η) = F (ξ , η) f (ξ , η), (3.7.1) with Kλ from (3.2.1). We already saw that (Kλ − E F (ξ , η))u(ξ , η) = (Aξλ + Aηλ )u(ξ , η). λ Now, using the completeness of the orthonormal base {φn,E }n∈N0 for E ∈ R, u possesses the expansion u(ξ , η) = ∑ n∈N0 where ψλn,E (ξ ) := ∫ λ unλ (ξ , η) with unλ (ξ , η) := φn,E (η)ψλn,E (ξ ), +π 0 (3.7.2) λ φn,E (η)u(ξ , η) sin η d η. This expansion extends to complex values of E by analyticity (note that no λ complex conjugate is involved, since φn,E is chosen real for E ∈ R). Analogously we get F (ξ , η) f (ξ , η) = ∑ n∈N0 where λ gn,E (ξ ) := ∫ +π 0 λ λ φn,E (η)gn,E (ξ ), λ φn,E (η)(F f )(ξ , η) sin η d η. (3.7.3) 3. THE TWO-CENTERS SYSTEM ON L2 (R3 ) 48 Substituting (3.7.2) and (3.7.3) into (3.7.1) one gets i∑ h ∑ λ λ (ξ ) Aξλ + Aηλ unλ (ξ , η) = φn,E (η) gn,E n∈N0 or equivalently ∑ n∈N0 n∈N0 λ λ φn,E (η) Anλ (E)ψλn,E (ξ ) − gn,E (ξ ) = 0. (3.7.4) If we can prove that (3.7.4) extends to complex points E ̸∈ σ(Hλ ) where Anλ (E) possesses an inverse Gnλ (E), we could rewrite the terms in (3.7.4) as λ ψλn,E (ξ ) = Gnλ (E) gn,E (ξ ) ∫ ∫ λ ˜ = Gn (ξ , ξ ; E) R+ (3.7.5) +π 0 λ φn,E (e η)(F f )(ξ˜, ηe) sin ηe sinh ξ˜ d ηe d ξ˜. Then, combining (3.7.5) and (3.7.2) we would obtain u(ξ , η) = ∑ n∈N0 λ φn,E (η) ∫∫ M0 λ Gnλ (ξ , ξ˜; E)φn,E (e η)(F f )(ξ˜, ηe) sin ηe sinh ξ˜ d ξ˜ d ηe and for each λ, we could read off the partial wave expansion for the Green function G λ (ξ , η; ξ˜, ηe; E) = ∑ n∈N0 λ λ (e η)Gnλ (ξ , ξ˜; E)(sinh2 ξ˜ + sin2 ηe) (η)φn,E φn,E (3.7.6) and try to analyze its convergence and its poles to define the resonances of our problem. The theory of Sturm-Liouville differential equations tells us that the inverse Gnλ (E) of Aλn (E) depends on special solutions of the equation, namely the “incoming and outgoing waves”. These are defined as solutions v± that decay exponentially for k ∈ C± , where k 2 = E and the solution v0 that is regular at the finite boundary point 0. If v+ (ξ ; E, n) (or v+ (ξ ; k, n)) extend analytically to the lower half-plane {ℑ k < 0}, then we can use the analyticity of any solution of the radial equation in ξ and some asymptotic estimates at ∞ to study the aforementioned Green’s function. It is completely unclear at this point how to study the asymptotic behavior of solutions of Anλ , but in view of Chapter 2 we may expect that the case λ = 0, corresponding to the planar situation, could be of help in suggesting how to proceed. Being in itself an interesting problem, we dedicate the next chapter to that topic. CHAPTER 4 The two-centers system on L2 (R2 ) Let us briefly recall the properties of the two-center coulomb equation in L2 (R2 ). It was already sketched in the previous chapter that it defines a self-adjoint operator bounded from below with form-domain H 1 (R2 ) (see [Kna89, Prop. 1] or [Agm82, Theorem 3.2]). 4.1. Separation of variables If we define, similarly as before, M := R+ × (−π, π), the restriction to M of the map ξ η 7→ cosh(ξ ) cos(η) sinh(ξ ) sin(η) defines a C ∞ diffeomorphism G : M → G(M ) whose image G(M ) is dense in R2 . Moreover it defines a change of coordinate from q ∈ R2 to (ξ , η) ∈ M , called prolate elliptic coordinates as in as (1.2.1). The classical picture and the characterization of the bifurcation digrams for this case coincide with the one for the planar case discussed in Section 2.3. The following result can be proven in the same way as Theorem 1.7. Theorem 4.1. Let ¦ © u ∈ Ca (R2 ) := C (R2 ) | u↾R2 \{±a} is twice continuously differentiable . The eigenvalue equation (−h 2 ∆ + V (q))u(q) = E u(q), E ∈ R, transformed to prolate elliptic coordinates, separates with the ansatz u ◦ G(ξ , η) = f (ξ ) g (η) into the decoupled system of ordinary differential equations 2 2 2 −h ∂ − Z cosh(ξ ) − E cosh (ξ ) + µ f (ξ ) = 0 + ξ −h 2 ∂η2 + Z− cos(η) + E cos2 (η) − µ g (η) = 0, where µ ∈ C is the separation constant, ¦ © f ∈ CN2 (R0+ ) := h ∈ C 2 (R0+ ) | h ′ (0) = 0 , ¦ © 2 g ∈ Cper ([−π, π]) := h ∈ C 2 ([−π, π]) | h (k) (−π) = h (k) (π) for k = 0, 1 49 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 50 and we have set Z± := Z2 ± Z1 and ∂α = ∂ . ∂α In fact, if F (ξ , η) is the function defined in (1.2.2) we have Proposition 4.2. The operator H on L2 (R2 ) defined as in (1.1.1) is unitarily equivalent to the operator in L2 (M , d χ ) given by HG := −h 2 ∆G + VG (ξ , η) where d χ := F (ξ , η) d ξ d η, ∆G := 1 F (ξ , η) ∂ξ2 + ∂η2 and VG (ξ , η) := − Z+ cosh(ξ ) − Z− cos(η) F (ξ , η) . HG has form core © ¦ G C0∞ (R2 ) = f (ξ , η) ∈ C0∞ M | f is 2π-periodic in η and ∂ξ f |ξ =0 = 0 and admits a unique self-adjoint realization with domain G (D(H )) where D(H ) is given by (4.1.1). Proof. The transformation to prolate elliptic coordinates (ξ , η) defines a unitary operator that we denote by G : L2 (R2 , d q) → L2 (M , d χ ). It is well known that H has a self-adjoint realization on L2 (R2 ) with domain ¦ © D(H ) := u ∈ L2 (R2 ) | V u ∈ L1loc (R2 ), u ∈ H 1 (R2 ), H u ∈ L2 (R2 ) , (4.1.1) where H u is to be understood in distributional sense (see [Agm82, Theorem 3.2] and Remark 3.2). The domain of the unitarily transformed HG := G H G −1 is then transformed to G (D(H )). Finally C0∞ (R2 ) is a form core for the quadratic form associated to H , therefore it is unitarily transformed to a form core for the quadratic form associated to HG . See [RS80, Section VIII.6] for the appropriate definitions. Then the form of the operator is given by Theorem 4.1. □ As discussed in Section 3.2 for the three dimensional case, we would like to move our point of view from the study of HG − E on L2 (M , d χ ) to the study of the separable operator KE := Kξ ⊗ 1l + 1l ⊗ Kη acting on L2 (M , d ξ d η) = L2 (R0+ , d ξ ) ⊗ L2 ([−π, π], d η). Here Kξ (h) := Kξ ,E,h := −h 2 ∂ξ2 − Z+ cosh(ξ ) − E cosh2 (ξ ), Kη (h) := Kη,E,h := −h 2 ∂η2 + Z− cos(η) + E cos2 (η). (4.1.2) 4.2. SPECTRUM OF THE ANGULAR EQUATION AND ITS ANALYTIC CONTINUATION 51 The separation reduces the problem to the study of two Sturm-Liouville equations (Kξ + µ) f (ξ ) = 0 and (Kη − µ) g (η) = 0 of the form (A.1) with p = r = 1 (see Theorem 4.1). As for the three dimensional problem, we call the first equation angular equation and the second equation radial equation. In particular they define essentially self-adjoint operators if we assume the proper boundary conditions on L2 (R0+ , d ξ ) and L2 ([−π, π], d η) respectively. More in details the eigenvalue equation of Kη (h) is in the class of the so called Hill’s equation. In view of Proposition 4.2, we are interested in the 2π-periodic solutions of the equation, i.e. we look for φ ∈ [−π, π] such that φ(−π) = φ(π) and φ ′ (−π) = φ ′ (π). For Kξ (h) it is clear that 0 is a regular point, we will see later how to treat the singular point ∞. For what concerns the boundary conditions in 0 we will require ϕ′ (0) = 0 (4.1.3) as suggested by Proposition 4.2. 4.2. Spectrum of the angular equation and its analytic continuation Let us shortly turn the attention to the angular equation. We will consider separately the case Z− = 0 and Z− ̸= 0. In the simplest case Z− = 0, Kη can be written in the form of a Mathieu equation −∂η2 ψ(η) − 2µ − E ψ(η) + 2 E cos(2η)ψ(η) = 0 (4.2.1) 4h 2 with periodic boundary conditions in [−π, π]. The potential V (η) := cos(2η) being 2h 2 even, it follows from Floquet theory (see [Tes00], [MW79], [MS54], [Eas75]), using the fundamental matrix F (α, δ) := f 1 f1′ f2 f2′ (π; λ, δ), λ := 2µ − E 2h 2 , δ := E 4h 2 , (4.2.2) built from the fundamental system f1 (0; λ, δ) = 1 = f2′ (0; λ, δ) , f2 (0; λ, δ) = 0 = f1′ (0; λ, δ), (4.2.3) that all the 2π-periodic solutions must be either π-periodic or π-antiperiodic in [0, π/2] (or [−π/2, 0]). In general the structure of the periodic solutions and their eigenvalues can be inferred by Theorem A.6. For the Mathieu equation, in particular, much more is known (see [MS54, Chapter 2]): for each integer n ≥ 0 we can find two solutions cen (x; δ) and sen+1 (x; δ) that have exactly n zeroes in (0, π) and that are π-periodic for even n and π-antiperiodic for odd n , the corresponding eigenvalues being λ+ (δ) n − − + and λn+1 (δ) respectively. For every δ ∈ R+ we have that λn and λn+1 are real and λ+ < λ+ < λ− < λ− < λ+ < ··· . 0 1 1 2 2 The following fact are proved in [Kat95, Chapter VII.3.3], [MS54, Chapter 2.2], [MSW80, Chapter 2.4] and [Vol98]. 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 52 (1) The eigenvalues of the Mathieu operators are analytic functions in δ with algebraic singularities. (2) The number of branch points is infinite, but countable, and there are no finite limit points. (δ) of δ by introducing (3) They can be defined uniquely as functions λ± n suitable cuts in the δ -plane, moreover they admit an expansion in powers of δ with finite convergence radius rn ≥ 2.04 n 2 . (4) For δ ∈ R no branch point is present and the eigenvalues are well defined real analytic functions of δ . (5) The number of branch points is infinite, but countable, and there are no finite limit points. (6) The operator T associated to (4.2.1) can be decomposed according to L2 (−π, π) = L+ ⊕ L+ ⊕ L− ⊕ L− 0 1 0 1 where the superscripts ± denote respectively the sets of even and odd functions and where the subscripts 0 and 1 denote respectively the sets of functions symmetric and antisymmetric with respect to x = π/2. (7) Each part of T is self-adjoint on the subset L± to which it is restricted and 01 has only simple eigenvalues as given by the following schema: L+ : λ0 , ϕ0 and λ+ , ϕ+ , 0 n n , , ϕ+ : λ+ L+ n n 1 L− : λ− , ϕ− , 0 n n L− 1 : λ− , ϕ− , n n n = 2, 4, 6, . . . ; n = 1, 3, 5, . . . ; n = 1, 3, 5, . . . ; n = 2, 4, 6, . . . . (8) All the eigenvalues in each of the four groups of the previous remark belong to the same analytic function, i.e. the eigenvalues in the same group lie on the same Riemann surface. (9) The eigenfunctions ϕ± (x) are called Mathieu Cosine and Mathieu Sine and n are themselves analytic functions of x and δ . Despite the completeness and the clarity of perturbation theory for one-parameter analytic families of self-adjoint operators, the situation is much more intricate and much less complete in presence of more parameters. And we cannot apply the theory as is (see Remark C.11). On the other hand we can use our restrictions on the parameters and the special symmetry of the potential to play in our favor. We know that Z− will be a real parameter, thus let us consider the equation −∂η2 ψ(η) + E − 2µ 2h 2 + Z− h2 cos(η) + E 2h 2 cos(2η) ψ(η) = 0 with periodic boundary conditions on [−π, π] where E ∈ R+ . Let us call λ := E − 2µ 2h 2 , γ1 = Z− h 2 , γ2 = E 2h 2 . (4.2.4) 4.2. SPECTRUM OF THE ANGULAR EQUATION AND ITS ANALYTIC CONTINUATION 53 Notice that the main difference between this case and the previous one is that now the period of the potential is no more smaller than the length of the considered interval, then by Floquet theory we look for solutions of period π instead of π/2. Remark 4.3. We already know from Theorem A.6 that for every choice of γ1 and γ2 the spectrum is discrete, at most double degenerate and accumulates at infinity. Anyhow it follows from [MW79, Theorem 7.10, p. 108] using a change of variable that in this case there cannot be coexistence of 2π-periodic eigenfunctions for the same eigenvalue. Thus the spectrum will be non-degenerate. In accord with the weaker result given in Theorem C.9, it is proved in [Str44] that, for E and Z− real valued, the eigenvalues are real and form a countably infinite set {λn (γ1 , γ2 )}n≥0 of transcendent real analytic (actually entire) functions of the parameters γ1 , γ2 ∈ R, so that in the (γ1 , γ2 , λ) space the sets γ1 , γ2 , λn (γ1 , γ2 ) | (γ1 , γ2 ) ∈ R define a countably infinite number of uniquely defined analytic surfaces. If we call Kη (E, Z− , h) the left hand side of (4.2.4), we can apply the results of Appendix C to T (β) := Kη (E, Z− , h) + β(1 + cos(2η)) where β is supposed to be defined by the real parameters Eim and h as follows β(Eim , h) := i Eim (with i = 2h 2 p −1). Therefore T (β) is nothing else that (4.2.4) with complex E . It is evident that T (β) defines a self-adjoint analytic family of type (A). Therefore it follows from Theorem C.7 and Remark C.10 that each λn (γ1 , γ2 , β) admits an analytic extension on the complex plane around each real E that can be expanded in series of Eim /2h 2 with an n -dependent convergence radius. Remark 4.3 concerning the simplicity of the spectrum and the construction described at point (7) keep to hold, therefore we may continue to regard each eigenvalue as simple restricted on its proper subspace and consider the lower bound of the convergence radius in terms of the eigenvalues spacing in the proper subspace. These distances are known to be at least of the order O(n) [Kat95, Chapter VII.2.4]. In the particular case considered, we can use the ansatz given by Theorem A.9 to bound the distance between the periodic solutions with a boundary value problem. To this end we can use the discussion of [Vol96, Section 5] and apply it to our case to obtain the following rough estimate generalizing point (3). Theorem 4.4. Let E > |2Z− |. Then the convergence radii ρD,N associated n with (4.2.4) with Dirichlet (resp. Neumann) boundary conditions satisfy lim inf n→∞ lim inf n→∞ ρn n2 ≥ ρn 6 13 (n − 1) 2 , ≥ 6 13 . 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 54 Proof. This is nothing else than Theorem 5.1 of [Vol96] where the estimate |2 cos(2x)| ≤ 2 (the Mathieu potential) is replaced by a corresponding estimate for Z cos(2x) + 2E− cos(x): if E ≫ |2Z− |, then Z− cos(x) ≤ 2. cos(2x) + 2 E □ Then the proof coincides with the proof in [Vol96] and the theorem follows. Remark 4.5. The estimates given in the previous theorem are underestimating the real convergence radius and could be improved upon, the main point for this section is to stress that the series expansion of the λn in term of the complex perturbation is likely to converge in a disk of radius of order O(n 2 ). 4.3. Asymptotic behavior of solution of the radial Schrödinger equation and their analytic extensions In this section we want to discuss some general estimates that we will need in order to justify the formal step in the separation of variables and the construction of the Green functions. We proceed similarly as in [AK92]. Consider the equation v ′′ (ξ , k) + h −2 k 2 cosh2 (ξ ) + Z+ cosh(ξ ) − µ v(ξ , k) = 0 (4.3.1) where ξ > 0, h > 0 and k ∈ C are arbitrary, and µ := µ l is the l th -eigenvalue of Aη for some l ∈ N (counted in ascending order for real parameters and then extended analytically). We assume w.l.o.g. that h = 1, since h can be absorbed in the other parameters. We will be interested in the solutions v± (ξ , k) := v± (ξ , k, µ) of (4.3.1) which decay as ξ → ∞ for k in the upper, resp. lower, half-plane C± = {ℑ(k) ≶ 0}. We call them, following [AK92] “outgoing”, resp. “incoming”, and we will make a specific choice of such a family of solutions by fixing the behavior of v± (ξ , k) as ξ → ∞. We want to construct a phase function that approximates the eikonal equation of the Schrödinger equation (4.3.1) up to a small error, that is characterized by a particular asymptotic behavior and that is analytic in k . We would like to consider something of the form ϕ(ξ , k) ∼ ∫ξq 0 k 2 cosh2 (t ) + Z+ cosh(t ) − µ d t . (4.3.2) Sadly this gives a well defined analytic function only for |k|2 > |Z+ − µ|. For our analysis it will be essential that the phase function is analytic in k ∈ C \ {0}. To construct it we reconsider the previous ansatz and perform a change of variables. If we call τ = sinh(t ), the above equation is transformed into ∫ ϕ(ξ , k) ∼ sinh(ξ ) Æ 0 k 2 − q(τ) d τ. with q(τ) := µ 1 + τ2 −p Z+ 1 + τ2 . If we call r = sinh(ξ ), we can imagine this ϕ(r ) as being the usual phase function asymptotic to k r (the phase function for the zero potential). 4.3. ANALYTICITY OF THE DECAYING SOLUTIONS 55 To start the construction of the proper phase function, we choose for every j ∈ N a decomposition q(τ) = s j (τ) + l j (τ), (4.3.3) where s j (τ) ∈ L1 (R+ ) and l j ∈ C 2 (R+ ) verifies sup l j (τ) ≤ 1/ j and r >0 l j (τ) = − p Z+ 1 + τ2 for τ > R j , where R j is some sequence of increasing positive numbers. Then the sequence of phase functions ∫ ϕ j (ξ , k) = sinh(ξ ) q 0 k 2 − l j (τ) d τ (4.3.4) is analytic in k ∈ Ω j := {k ∈ C | |k|2 > 1/ j } and of class C 2 in ξ ∈ R+ . In (4.3.4) we have taken the principal branch of the p square root, i.e. the uniquely determined analytic branch of z that maps R+ into itself. We want to show now that these analytic functions can be modified by analytic functions of k alone such that they all patch together to define a “global” phase function ϕ(ξ , k) which is analytic in k ∈ C\{0} and assigns to each k a germ of a function of ξ at infinity. To this end we want to make use of the following theorem (see [Hör73, Theorem 1.4.3]). Theorem 4.6 (Mittag-Leffler Theorem). Let Ω = ∪ j Ω j , where the Ω j are open sets in C, and let there be given meromorphic functions g j , respectively, on the sets Ω j , where the differences g j − gk are analytic functions on the intersections Ω j ∩ Ωk for all j and k . Then there is on Ω a meromorphic function f such that the differences f − g j are analytic on Ω j for all j . In fact we have the following theorem. Theorem 4.7. There is a global phase function ϕ(ξ , k) such that for all j ∈ N there is a g j in the space A(Ω j ) of analytic functions in Ω j , such that ϕ(ξ , k) = ϕ j (ξ , k) + g j (k) e , k ∈Ω , for ξ > R j j (4.3.5) and ϕ(ξ , −k) = −ϕ(ξ , k) for k ∈ C \ {0}. (4.3.6) Proof. It follows from the definition given in (4.3.4) that the function gi l (k) = ϕ j (r, k) − ϕ l (r, k) for r > arcsinh(Rmax{ j ,l } ) (4.3.7) is analytic in k ∈ Ωmin{ j ,l } . By definition, these functions satisfy the cocycle condition g j l = −g l j , g j l + g l h + g h j = 0 in Ω j ∩Ω l ∩Ω h , ∀ j , k, h ∈ N. (4.3.8) 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 56 In view of (4.3.8), we can remove an index using the Mittag-Leffler Theorem 4.6. In fact it implies that for all j ∈ N there is g j ∈ A(Ω j ) such that g j l = gl − g j in Ω j ∩ Ω l . (4.3.9) Since ϕ j (r, −k) = −ϕ j (r, k), it follows from (4.3.9) and (4.3.7) that 1 1 ϕ j (ξ , k) + ( g j (k) − g j (−k)) = ϕ l (ξ , k) + ( g l (k) − g l (−k)) 2 2 in the common domain of definition. Thus the local data ϕ j + 12 ( g j (k) − g j (−k)) patch together to define a global phase function satisfying (4.3.5) and (4.3.6). □ Henceforth we will refer to the ϕ(ξ , k) defined in Theorem 4.7 as a global phase function. Remark 4.8. Choosing in (4.3.3) a different decomposition of q(τ) into a short range and long range part, keeping l (ξ ) fixed near infinity, modifies ϕ(ξ , k) µ by an analytic function of k alone. In particular the term s (τ) := being in the 1+τ 2 short range component s j of (4.3.3) will be dropped out for the moment. Proposition 4.9. The phase function ϕ(ξ , k) has the asymptotic behavior given by ϕ(ξ , k) = k sinh(ξ ) + Z+ 2k ξ + O(1) = k 2 e ξ (1 + o(1)) as ξ → ∞. (4.3.10) Proof. Without losing generality we can suppose |k| > |Z+ | and consider the phase function given by ϕ(ξ , k) = ∫ξq 0 ∫ =k ξ k 2 cosh2 (t ) + Z+ cosh(t ) d t s Z+ cosh(t ) 1 + k 2 cosh(t ) 0 d t. We are interested in the asymptotic behavior of ϕ(ξ , k) as ξ → ∞. Using ∫ ϕ(ξ , k) = k ξ cosh(t ) 1 + 0 = k sinh(ξ ) + Z+ 2k ξ Z+ 2k 2 cosh(t ) ξ− Z+2 8k 3 +O k −2 cosh (t ) dt −2 gd(ξ ) + O(1), where gd(x) := arctan tanh 2 is the Gudermannian function. Writing sinh(ξ ) = x −x (e − e )/2 and collecting the growing term we have the thesis. □ Remark 4.10. Notice that all the terms appearing in the O(1) above can be explicitly computed and bounded in terms of a function of k and Z+ . The Liouville-Green Theorem B.17 guarantees that for each k ∈ C there exist two linearly independent solutions of (4.3.1) such that y1,2 (ξ ) = Æ 1 ′ ϕ (ξ , k) e ±i ϕ(x,k) (1 + o(1)) for ξ → ∞. 4.3. ANALYTICITY OF THE DECAYING SOLUTIONS 57 In particular, it follows from the asymptotic estimate of Proposition 4.9 that (4.3.1) must be in the Limit Point Case at infinity – Case I – of Theorem B.3 if we set r (x) := cosh2 (x), p := 1 and λ := k 2 . In what follows we investigate the regularity of the solutions with respect to ξ and k . Theorem 4.11. For each k ∈ C \ {0}, equation (4.3.1) has a unique solution v± (ξ , k) verifying the asymptotic relation p ξ v± (ξ , k) = 2e − 2 e ±i ϕ(ξ ,k) (1 + o(1)) as ξ → ∞, (4.3.11) where (4.3.11) holds uniformly in any truncated angle Λ± (η, δ) = {k ∈ C \ {0} | η ≤ arg(±k) ≤ π − η, |k| ≥ δ} with η ≥ 0, δ > 0. The family of solutions defined by (4.3.11) is analytic in k ∈ C± \ {0} := {z ∈ C \ {0} | ±ℑz > 0} and extends continuously to k ∈ C± \ {0}. Remark 4.12. (4.3.6) and the uniqueness of Theorem 4.11 imply that v+ (ξ , k) = v− (ξ , −k). In particular it suffices to consider v+ . Proof. In view of Theorem 4.7 and the subsequent remark, we can reduce the proof to the case where the phase function is given by ϕ(ξ , k) = ∫ξq 0 k 2 cosh2 (t ) + Z+ cosh(t ) d t (4.3.12) for ξ > 0 and |k|2 > |Z+ |. We call this ϕ(ξ , k) a local phase function. Let V± (ξ , k) := 1 k 2 ϕ′ (ξ , k) e ±i ϕ(ξ ,k) (4.3.13) define the approximate solutions of (4.3.1). For |k| ≥ δ the function V± satisfies the comparison equation V±′′ (ξ , k) + k cosh (ξ ) + Z+ cosh(ξ ) + Sϕ(ξ , k) V± (ξ , k) = 0 2 1 2 2 (4.3.14) where Sϕ denote the Schwarzian derivative Sϕ = ϕ′′′ ϕ′ − 3 2 ϕ′′ 2 ϕ′ . (4.3.15) For k ∈ Λ± (η, δ) the inhomogeneous Volterra Integral Equation [Tri85] ∫ v± (ξ , k) = V± (ξ , k) − ∞ ξ Kk (ξ , t )Fk (t )v± (t , k) d t (4.3.16) where Fk (t ) = 12 Sϕ(t , k) + µ is the function that expresses the difference between the Schrödinger equation (4.3.1) and the comparison equation (4.3.14) and K(ξ , t ) is the Green’s function associated with equation (4.3.13): Kk (ξ , t ) = W (V− ,V+ )−1 V+ (ξ )V− (t ) − V+ (t )V− (ξ ) , with Wronskian W (V− ,V+ ) := V−V+′ − V−′ V+ = 2i k . (4.3.17) 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 58 To give (4.3.16) meaning we need to check if the definition makes sense and a solution can be found. From now one we may suppress writing the additional parameter k whenever it is not necessary. We can explicitly compute F (t ) using (4.3.15) obtaining 10k 4 − Z+2 − 2k 4 cosh(2ξ ) + Z+ sech(ξ ) 12k 2 + 5Z+ sech(ξ ) Sϕ(ξ ) = 2 8 Z+ + k 2 cosh(ξ ) and thus, for real ξ and for every k ∈ Λ+ (η, δ), we have lim |F (ξ )| = ξ →∞ 1 8 +µ and sup |F (ξ )| =: CF < ∞. (4.3.18) ξ ∈R+ Of course CF depends on Z+ , µ and k , thus on η and δ . Moreover from (4.3.13) and (4.3.10), writing k ∈ Λ+ (η, δ) as k = k r + i ki (k r , ki real), we get ϕ(ξ ,k) k p −ξ ξ V (ξ , k) = 2e 2 (1 + o(1)) e i k k ≤ C e − 2 e − 2i e ξ (1+o(1)) , V ± where CV (k) := sup ξ ∈R+ (4.3.19) e ξ /2 |k/ϕ′ (ξ , k)| < ∞. Therefore for 0 < ξ ≤ t < ∞ we have v u 2 1 u k t i (ϕ(ξ ,k)−ϕ(t ,k)) i (ϕ(t ,k)−ϕ(ξ ,k)) |K(ξ , t )| = e − e 2i k ϕ′ (t , k)ϕ′ (ξ , k) Ç CV2 − ξ +t ∫t Z+ 2 ≤ e exp −i k ξ cosh(τ) 1 + k 2 cosh(τ) d τ 2 Ç ∫t Z − exp i k ξ cosh(τ) 1 + 2 + d τ k cosh(τ) Ç CV2 − ξ +t ∫t Z+ 2 ≤ e CK exp −i k ξ cosh(τ) 1 + 2 d τ , (4.3.20) k cosh(τ) 2 where Ç ∫t Z+ CK (k) := sup 1 − exp 2i k ξ cosh(τ) 1 + 2 d τ ≤ 2. k cosh(τ) + t ,ξ ∈R It follows from (4.3.18), (4.3.19) and (4.3.20) that the Volterra Integral Equation (4.3.16) is well defined as a mapping from the function space n C± (η, δ) := f ∈ C 2 R+ × Λ± (η, δ) ∀k ∈ Λ± (η, δ), o ∥ f ∥k := sup f (x, k) e ∓iϕ(x,k) < ∞ x∈R+ (4.3.21) to itself. In particular, being V± ∈ C± (η, δ) we can apply the Picard iteration procedure to find a solution of the equation and prove its existence. We claim that 4.3. ANALYTICITY OF THE DECAYING SOLUTIONS 59 the solution must be unique. Suppose there exists two solutions v+ , w+ ∈ C+ of (4.3.16), then ∫ ψ(ξ , k) := v+ (ξ , k) − w+ (ξ , k) = − ∞ ξ K(ξ , t )F (t )ψ(t , k) d t . (4.3.22) The previous estimates applied to (4.3.22) give ∫ ∞ |ψ(ξ , k)| = K(ξ , t )F (t )ψ(t , k) d t ξ ∫∞ |K(ξ , t )F (t )ψ(t , k)| d t ≤ ξ v u 2 k t ≤ ′ ′ 2 ϕ (t , k)ϕ (ξ , k) ξ Ç ∫t Z + · exp −i k ξ cosh(τ) 1 + 2 d τ e i ϕ(t ,k) d t k cosh(τ) s ∫ s CK CF Cψ i ϕ(ξ ,k) ∞ k k ≤ ϕ′ (t , k) d t ϕ′ (ξ , k) e 2 ξ ∫ ∞ p CK CF Cψ CV ξ t ≤ e − 2 e i ϕ(ξ ,k) 2e − 2 (1 + o(1)) d t 2 ξ ∫ ∞ t CK CF Cψ CV CI ξ −2 i ϕ(ξ ,k) ≤ e e e− 2 d t 2 ξ −ξ i ϕ(ξ ,k) = Cψ C t ot e e (4.3.23) CK C F Cψ ∫ ∞ u where Cψ (k) := ∥ψ∥k , CI := supξ ∈R + Ç p −2ξ 2 (1 + e ) 1+ Z+ k 2 cosh(ξ ) − 1 2 and C t ot := CK CF CV CI . Using equations (4.3.22) and (4.3.23)we can reiterate the procedure, in fact defining ∫ ψ1 (ξ , k) := ψn (ξ , k) := ∞ ξ ∫∞ ξ K(ξ , t )F (t )ψ(t , k) d t , K(ξ , t )F (t )ψn−1 (t , k) d t , one can prove by induction that |ψ(ξ , k)| = |ψn (ξ , k)| ≤ C tnot e −nξ (2n − 1)(2n − 3) · · · 3 · 1 C n e −nξ i ϕ(ξ ,k) ≤ Cψ t ot e n! iϕ(ξ ,k) e (4.3.24) 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 60 uniformly in k ∈ Λ+ (η, δ) and for all n ∈ N. The convergence of ∞ ∑ n=1 Cψ C tnot n! −ξ e −nξ e iϕ(ξ ,k) = Cψ e iϕ(ξ ,k) e Ct ot e − 1 (4.3.25) implies that |ψ(ξ , k)| = 0, i.e. w+ = v+ . The same inequality implies that after some iterates the homogeneous integral equation is a contraction and coupled with the bounds on V+ it follows that (4.3.16) has a unique fix point. Thus proving the existence and uniqueness of the solution. In fact if we define v0,+ (ξ , k) := V+ (ξ , k), ∫∞ vn,+ (ξ , k) := − K(ξ , t )F (t )vn−1,+ (t , k) d t , ξ ∑∞ v , the series converges absolutely n=0 n,+ −ξ uniformly in k ∈ Λ+ (η, δ) with v+ (ξ , k) ≤ V+ (ξ , k) e C e for some positive constant C . Therefore one has the Picard iteration converges to v+ = v+ (ξ , k) = V+ (ξ , k)(1 + o(1)) as ξ →∞ and (4.3.11) holds. The fact that all the bounds are valid for k ∈ R completes the proof. □ Remark 4.13. It is possible to compute an explicit bound like (4.3.24) using the fact that C n e −nξ ξ t ot . vn,+ ≤ CV e − 2 e i ϕ(ξ ,k) 2n n! In particular the dependence on the small-range parameter µ appear in the C t ot and in view of the previous estimate can be bounded by |µ|O(1), therefore we can be more precise and say that v± (ξ , k) = p ξ 2e − 2 e ±i ϕ(ξ ,k) 1 + M± (ξ , kµ) as ξ → ∞, (4.3.26) where for some constant C ̸= 0 we have −ξ M± (ξ , k, µ) = e C |µ|e o(1). Remark 4.14. Let ve any other family of solutions of (4.3.1), analytic in k ∈ C \ {0} and satisfying for k ∈ Λ+ (η, δ) e , k) = o(1) v(ξ as ξ → ∞. Then e , k) = γ (k)v+ (ξ , k), v(ξ where γ (k) is a nowhere-vanishing analytic function of k ∈ Λ+ (η, δ). 4.3. ANALYTICITY OF THE DECAYING SOLUTIONS 61 In the notation of Appendix B we set α = π2 , r (x) = cosh2 (x), λ = k 2 , p(x) = 1 and q(x) = µ − Z+ cosh(x). Let µ ∈ C− , then we have for Q from (B.1) ( Q⊂ ) ℜµ − Z+ cosh(ξ ) z ∈ C ℜz ≥ inf , ℑµ ≤ ℑz ≤ 0 ξ ≥0 cosh2 (ξ ) and Q b = R+ for Q b defined in (B.11). Therefore by Theorem B.9 we have that m(k 2 ) is defined throughout C \ Q , that has a unique connected component, and has a meromorphic extension to C \ R+ with poles only in Q \ R+ . Let k 2 ∈ C \ Q , being v+ (ξ , k) the only integrable solution, it must be up to a constant defined by v+ (ξ , k) = θ(ξ , k 2 ) + m(k 2 )ϕ(ξ , k 2 ). The fact that the two solution coincide for k in an open subset of the complex plane implies that they must represent the same analytic function, confirming in particular our Theorem 4.7. Notice that in particular this means that m(k 2 ) = M (k) is a function of the square root of the actual eigenvalue k 2 . For k 2 ∈ C \ Q we can define the Green function as in (B.12) ( −ϕ(ξ , k 2 )ψ(ξe, k 2 ), 0 < ξ < ξe < ∞ G(ξ , ξe; k ) = −ψ(ξ , k 2 )ϕ(ξe, k 2 ), 0 < ξe < ξ < ∞ (4.3.27) ( 1 −ϕ(ξ , k 2 )v+ (ξe, k), 0 < ξ < ξe < ∞ = [ϕ, v+ ] −v+ (ξ , k)ϕ(ξe, k 2 ), 0 < ξe < ξ < ∞ 2 and the resolvent Rλ as in (B.13). Remark 4.15. In case Z+ = 0, the solutions of (4.3.1) are given by linear combinations of the modified Mathieu functions (Mc and Sc) [EMOT55]. In particular, if we look at their asymptotic behavior, we find out that up to a constant factor k2 k2 v+ (ξ , k) = Mc µ − , , ξ 2 4 ! (4.3.28) where Mc(a, q, x) is the modified Mathieu cosine, i.e. the solution of y ′′ (x) + (a − 2q cosh(2x))y(x) = 0 p that decays for q ∈ C+ . It is well known [MS54, Chapter 2] that the function in the RHS of (4.3.28) admits an analytic continuation through the positive real axis on the negative complex plane for −π/2 ≤ arg(k) ≤ π/2 and that for x → ∞ and k ∈ C+ it has the following asymptotic behavior [MS54, EMOT55] k2 k2 Mc µ − , , x 2 4 ! x k x (1+o(1)) = e− 2 ei 2 e (1 + o(1)), in line with what we expected. For what follows we will need to work in a slightly different setting. If we perform the change of variable defined by ξ 7→ log(x + 1) where we consider the principal 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 62 branch of the logarithm in case of continuation to the complex plane, equation (4.3.1) takes the form ′ (x + 1)w ′ (x, k) + + 1 k2 h2 4 1 h2 x + 1 + (x + 1)−1 + (x + 1)−3 Z+ 2 −2 1 + (x + 1) ! w(x, k) −1 − µ(x + 1) w(x, k) = 0 (4.3.29) where x > 0, h > 0 and k ∈ C \ {0}. As before we assume h = 1 for the moment. Remark 4.16. In this case Theorem 4.11 and Remark 4.14 is still valid and in accord with the Liouville-Green Theorem B.17 we have two unique solutions that as x → ∞ are asymptotic to w± (x, k) = p =p 1 x +1 1 e e ±i ϕ(x,k) (1 + o(1)) Z exp ±i k2 x + 2k+ log(x + 1) + k2 x +1 2 Z+ −1 −2 · exp ±i + O (x + 1) (1 + o(1)) 3 (x + 1) (4.3.30) 4k e k) = ϕ(log(x +1), k). The asymptotic behavior (4.3.30) holds uniformly where ϕ(x, for k in any sector Λ± (η, δ) = {k ∈ C | η ≤ arg(±k) ≤ π − η, |k| ≥ δ} with η ≥ 0 and δ > 0. The family of solutions defined by (4.3.30) is analytic in k ∈ C± \ {0} and extends continuously to k ∈ C± \ {0}. Remark 4.17. From now on we write with an abuse of notation ϕ(x, k) in e k). place of ϕ(x, Before proving the main theorem we need the following lemma. Lemma 4.18. Let K be a compact set in C \ {0}. Then for any −π < θ < π, there is a constant Aθ such that any solution of equation (4.3.29) verifies the estimate 1 |v(x, k)| ≤ Aθ (|c| + |c ′ |) p e |ℑϕ(x,k)| x (4.3.31) for x ∈ e i θ [1, ∞) and k ∈ K , where c = v(0, k), c ′ = v ′ (0, k), (4.3.32) denote the initial data at x = 0. Proof. We start proving (4.3.31) in the case η ≤ | arg k| ≤ π − η for any η ≥ 0 and θ = 0 (i.e. x ∈ R+ ). All the constants that we are going to use without an explicit definition are defined as previously. Using the approximate solutions given 4.3. ANALYTICITY OF THE DECAYING SOLUTIONS 63 by (4.3.13) defined with an abuse of notation by V± (x, k) := V± (log(x + 1), k), we determine a+ and a− from the initial data requiring c = a+V+ (0, k) + a−V− (0, k), (4.3.33) c ′ = a+V+′ (0, k) + a−V−′ (0, k). Then v(x, k) satisfies the Volterra Integral Equation ∫ v(x, k) = a+V+ (x, k) + a−V− (x, k) + x K(x, t )F (t )v(t , k) 0 dt t +1 (4.3.34) where K(x, t ) := K(log(x + 1), log(t + 1)) and F (t ) := F (log(t + 1)) are defined with an abuse of notation from the respective function (4.3.17) and (4.3.16). Notice similarly as in the previous theorem that for 0 ≤ t ≤ x there exist a constant C0 (η, δ) such that we have C0 (η, δ) 1 1 |ℑ(ϕ(x,k)−ϕ(t ,k))| |K(x, k)| ≤ e ′ ϕ (x, k) ϕ′ (t , k) 2 ≤ CV2 C0 (η, δ) 2 p 1 (x + 1)(t + 1) e |ℑ(ϕ(x,k)−ϕ(t ,k))| . (4.3.35) Define now V (x, k) = p 2(|a+ | + |a− |) p 1 x +1 e |ℑϕ(x,k)| . (4.3.36) The sequence v0 (x, k) := a+V+ (x, k) + a−V− (x, k), ∫x dt , vn (x, k) := K(x, t )F (t )vn−1 (t , k) t +1 0 is uniformly convergent. In fact, suppressing the dependence of the constant on η and δ , we have |v0 (x, k)| ≤ CV V (x, k) and, using the transformed version of (4.3.35), it follows by induction that |vn (x, k)| ≤ where 1 n! V (x, k)Ln (x), (4.3.37) ∫ x dt 1 L(x) := C0 ′ |F (t )| t +1 0 ϕ (t , k) ∫x dt 1 |F (t )| = C0 CV p t +1 t +1 0 1 ≤ C0 CV CF p x +1 ∑ is uniformly bounded for x ∈ R+ . Therefore ∞ v (x, k) converges uniformly and n=0 n absolutely and coincides with the given solution v(x, k) of (4.3.34) for η ≤ | arg k| ≤ π − η, η ≥ 0. In particular being a± bounded in terms of the initial data c and c ′ , we obtain (4.3.31) for real values of x . 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 64 At this point it is enough to notice that as soon as we do not cross the branch cut of the logarithm all the inequalities and the equations written up to this point are valid, therefore the result holds replacing x with e i θ x for every−π < θ < π. □ We are ready to prove that the functions v± can be analytically extended in k up to the positive real axis, to this end we consider the transformed w± . Remark 4.19. The potential is analytic in C \ (−∞, −1], therefore its analyticity in the cone Σα,β := {z ∈ C | −α < arg z < β, |z| > 0} (4.3.38) for some α, β ∈ [0, π) is clear. Theorem 4.20. Let w± (x, k) be defined as in Remark 4.16. Then w+ (x, k) admits an analytic continuation in k through the positive real k -axis into the region {k ∈ C \ {0} | −β < arg k < β} , w− (x, k) admits an analytic continuation into {k ∈ C \ {0} | −α < arg k < α} , for any α, β ∈ [0, π) and both verify the asymptotic relation (4.3.11) 1 w± (x, k) = p e ±i ϕ(x,k) (1 + o(1)) x as x → ∞ in Σα,β , (4.3.39) where (4.3.39) holds locally uniformly in k and uniformly in x . Furthermore an analytic continuation of w+ (x, k) and w− (x, k) through the negative real axis is defined via w+ (x, k) = w− (x, −k). (4.3.40) Remark 4.21. If α+β > π, the analytically continued function w± (x, k) may be double valued for k ∈ Z∓ . By an abuse of notation we denote the corresponding possibly self-overlapping domain by D± (α, β) := {k ∈ C \ {0} | −β < arg(±k) < π + α} (4.3.41) Proof. It is well known [CL55, Chapter 3.7] that, as solutions of the linear differential equation (4.3.29) with analytic coefficients, w± (x, k) admit an analytic continuation in x into the region Σα,β . The main point of this proof is to use this information to obtain the analyticity in k via dilation. More in details we plan to resemble the strategy of [AK92, Theorem 2.6] refining the crude bound of Theorem 4.18 using the Phragmen-Lindelöf principle. This allows us to identify the dilated solutions with a decaying solution of the dilated equation. In view of Lemma 4.18, this solution is uniquely defined by the asymptotic behavior as x goes to infinity. Let us consider w+ (z, k) along a ray Γ := {z | arg z = γ } with 0 < γ < β. Then for x > 1 and k ∈ C+ \ {0}, the function ω(x, k, γ ) = w+ (e i γ x, k) (4.3.42) 4.3. ANALYTICITY OF THE DECAYING SOLUTIONS 65 satisfies the equation ′ e 2i γ (e i γ x + 1)ω ′ (x, k) + 2 q(e i γ x, k, Z+ , µ)ω(x, k) = 0 h (4.3.43) where k2 (x + 1) + (x + 1)−1 + (x + 1)−3 4 Z+ + 1 + (x + 1)−2 − µ(x + 1)−1 . 2 q(x, k, Z+ , µ) := Moreover the initial data ω ′ (0, k, γ ) = e i γ w+′ (0, k), ω(0, k, γ ) = w+ (0, k), (4.3.44) are analytic in k ∈ C+ \ {0}. To obtain an analytic continuation of w+ (x, k) into the lower half-plane first observe that by the Liouville-Green Theorem B.17 and Remark 4.16, equation (4.3.43) has a unique solution ω+ (x, k, γ ) in the half-plane −γ < arg k < π−γ characterized by the asymptotic relation ω+ (x, k, γ ) := p 1 iγ e i ϕ(e iγ x,k) (1 + o(1)) as x → ∞. (4.3.45) e x We claim that in fact ω+ (x, k, γ ) = ω(x, k, γ ) for x ∈ R+ , 0 < arg k < π − γ . (4.3.46) ′ Then ω+ (1, k, γ ) and ω+ (1, k, γ ) provide the analytic continuation of the initial data for w+ (x, k) into the region −γ < arg k < 0, implying that w+ (x, k) can be continued analytically into the lower half-plane. To prove (4.3.46), we observe that x 7→ w+ (x, k) is of exponential type for x ∈ Σα,β and decays exponentially for ℑk > 0. Then it follows from the PhragmenLindelöf principle [Con78, VI.4] that for fixed ℑk > 0 the function w+ (x, k) decays exponentially as x → ∞ in a small cone containing R+ . Therefore Remark 4.16 and Remark 4.14 applied to the dilated function ω+ (x, k, γe) for some small γe > 0 imply that ω+ (x, k, γe) is a multiple of ω(x, k, γe). This means moreover that it decays at a rate given by the expected function p 1 i γe e i ϕ(e i γe x,k) . e x We can repeat this procedure a finite number of times and deduce that for fixed k the analytic function g (x, k) := p x e −iϕ(x,k) w+ (x, k) (4.3.47) is uniformly bounded as x → ∞ within an angle −ε < arg x < γ + ε for some ε > 0. Since lim g (x, k) = 1, x→∞ it follows from Montel’s theorem [Con78, VII.2] that this limit is assumed uniformly as x → ∞ in 0 ≤ arg x ≤ γ . This proves (4.3.46). Since γ ∈ (0, β) was arbitrary, 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 66 we obtain an analytic continuation of w+ (x, k) to −β < arg k < π. It remains to prove (4.3.39). For −α < γ < β we can apply Lemma 4.18 to the dilated function ω(x, k, γ ) to have g (x, k) = O(1) as x → ∞ within Σα,β . (4.3.48) We already know from (4.3.46) that g (x, k) → 1 as x → ∞ along any ray such that 0 < η ≤ arg(k x) ≤ π − η for some η ≥ 0. Therefore we have that locally uniformly for k ∈ C \ {0}, −β < arg k < π g (x, k) = O(1) as x → ∞ within Σα,β and g (x, k) is uniformly bounded along the boundary rays of Σα,β . That g (x, k) is uniformly bounded in x ∈ Σα,β is now a consequence of the Phragmen-Lindelöf Principle. The fact that g (x, k) tends to 1 as x → ∞ since it does so along some ray contained in its interior, completes the proof of the theorem. □ 4.4. Generalized eigenfunctions, Green’s function and the scattering matrix We are now ready to construct the main elements for the partial wave expansion required to give a definition of the resonances of our operator. We considered in the previous section the incoming and outgoing waves as the “regular” solutions at infinity, and because of the fact that they are defined around infinity it required some work to prove that they can be analytically extended to the second Riemann sheet across the positive real axis. The situation is completely different if we consider the solution w0 (x, k) of (4.3.29) (or the corresponding v0 (ξ , k) of 4.3.1) that is regular in 0 in the sense of the boundary conditions derived from (4.1.3), i.e. w0′ (0, k) = 0. (4.4.1) For the moment we can assume that w0 is normalized so that w0 (0, k) = 1. Being the solution of a boundary problem with analytic coefficients and analytic initial conditions, the following theorem follows as a corollary of the standard theory of complex ordinary differential equations (see [CL55, Chapter 1.8]). Theorem 4.22. The unique solution w0 (x, k) of (4.3.29) defined by the condition (4.4.1) is analytic in x ∈ Σα,β , k ∈ C \ {0} and satisfies w0 (x, k) = w0 (x, −k). (4.4.2) Remark 4.23. Working with (4.3.29) or (4.3.1) is equivalent. We will use each time the representation that makes the proofs and the computations easier. Therefore in what follows we do not continue to remark that the properties are equivalent. It is always possible to understand in which setting we are working, looking at the name of the functions and the variables. From now on, we will always assume that the Wronskian is defined it is generalized form given by (A.3). 4.4. SCATTERING ON THE LINE 67 We are finally ready to introduce the basic elements for scattering theory on the half-line. We call Jost functions associated to the radial equation (4.3.29) and our choice of phase function ϕ(x, k) the Wronskians f± (k) := W (w± (x, k), w0 (x, k)). (4.4.3) They connect the regular solution to the incoming and outgoing ones via the identity W (w− , w+ )w0 = f+ w− − f− w+ , W (w+ , w− ) = 2i k, (4.4.4) that follows expanding explicitly the Wronskian and using the asymptotic behavior of the solutions in their domain of analyticity. In particular this implies the following corollary of Theorem 4.22 and Theorem 4.20. Corollary 4.24. The Jost functions f± (k) are analytic in k ∈ D± (α, β) defined in (4.3.41) and verify p f± (k) = ±(2i k) lim e iγ /2 x exp ±i ϕ(e i γ x, k) w0 (e i γ x, k), x→∞ (4.4.5) where γ ∈ (−α, β) satisfies γ ≷ − arg(k) according to the choice of sign of (4.4.5). It will be convenient for what follows to change the normalization w0 (0, k) = 1 to one at “infinity” in the sense of Corollary 4.24. Namely if f+ (k) ̸= 0, we define the generalized eigenfunction of the radial equation (4.3.29) and our choice of phase function ϕ(x, k) the function e(x, k) := f+ (k)−1 w0 (x, k). (4.4.6) With this notation we introduce for k ∈ Σα,β with f+ (k) ̸= 0 the radial Green’s function G(x, x ′ ; k) := e(x< , k)w+ (x> , k), (4.4.7) where for x, x ′ > 0, x< = min{x, x ′ } and x> = max{x, x ′ }. G(x, x ′ ; k) is a fundamental solution of the radial Schrödinger equation (4.3.29). In the appropriate coordinate representation it coincides for ℑk > 0 with (4.3.27) and therefore with the resolvent kernel (B.12) of the (Z+ , µ)-dependent J -self-adjoint realization of the radial Schrödinger operator associated to the Sturm-Liouville equation (4.3.1) in L2 (R+ , cosh2 (ξ )d ξ ), described by Theorem B.15. Remark 4.25. We now consider the spectral parameter µ appearing in Equation (4.3.1) as a perturbation of the operator Kξ defined in (4.1.2). Consequently we will write Kξ (Z+ , µ) := Kξ + µ for the perturbed operator. Remark 4.26. Notice that eventual zeros of f+ (k) for k ∈ C+ \{0} correspond to eigenvalues of the operator. 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 68 In view of Theorem 4.20 and 4.22, G(x, x ′ ; k) possesses a meromorphic continuation in k into the possibly two-sheeted domain D+ (α, β) defined by (4.3.41). Finally we introduce the so-called scattering matrix element s (k) = f− (k) (4.4.8) f+ (k) which in lieu of Corollary 4.24 is a meromorphic function of k in D+ (α, β)∩D− (α, β). Lemma 4.27. Let x, x ′ > 0 and −β < arg(k) < α. (1) The radial Green’s function and the radial generalized eigenfunctions satisfy the functional relation G(x, x ′ ; k) − G(x, x ′ ; −k) = 2i k e(x, k)e(x, −k). (4.4.9) (2) The scattering matrix element satisfy the following relation s (−k) = s (k)−1 . (4.4.10) (3) The scattering matrix elements and the radial generalized eigenfunctions satisfy the functional relation s (k)e(x, −k) = e(x, k). (4.4.11) Proof. From (4.3.40) and (4.4.2) we have that f+ (−k) = f− (k) (4.4.12) for k ∈ D+ (α, β) ∩ D− (α, β). Therefore, using (4.4.4) and the definitions of the radial Green’s function and the radial generalized eigenfunctions the first part of the lemma is proved. The second part and the third part follows as a direct application of (4.4.12) to the definition of the scattering matrix elements. □ A first consequence of Lemma 4.27 is that it is enough to discuss the scattering matrix elements in the angle −β < arg(k) < α. With the above definitions we can discuss the notion of eigenvalues for the radial non-selfadjoint Schrödinger operator Kξ (Z+ , µ) in L2 (R+ , cosh2 (ξ )d ξ ). We define n EZ+ ,µ := k ∈ C+ \ {0} | f+ (k) = 0, o e −ξ /2 e i ϕ(ξ ,k) ∈ L2 (R+ , cosh2 (ξ )d ξ ) . (4.4.13) If k ∈ E , we call k an eigenvalue of this quadratic eigenvalue problem. All other zeros of the Jost function f+ (k) are called resonances of Kξ (Z+ , µ) and we denote them by RZ+ ,µ := k ∈ D+ (α, β) \ E | f+ (k) = 0 . (4.4.14) Remark 4.28. The condition e i ϕ(ξ ,k) ∈ L2 (R+ , cosh2 (ξ )d ξ ) is automatically fulfilled when k ∈ C+ \ {0} and for it is independent of µ. 4.5. FORMAL “PARTIAL WAVE EXPANSION” OF THE GREEN’S FUNCTION 69 Remark 4.29. There cannot be real positive k ∈ EZ+ ,µ . In fact, if there exists k ∈ R+ in EZ+ ,µ , then by Theorem 4.20 we would have v+ (x, k) ∈ L2 (R+ , cosh2 (ξ )d ξ ) but it is evident from the asymptotic behavior of v+ that this is impossible. On the other hand, we cannot exclude a priori the presence of real k in RZ ,µ . + Remark 4.30. Two Jost functions cannot vanish simultaneously in −β < arg(k) < α, otherwise v+ and v− (or w+ and w− ) would be linearly dependent in contradiction with their asymptotic behavior. Therefore the points of EZ ,µ ∪ R + contained in −β < arg(k) < α are in one to one correspondence with all the poles of the scattering matrix elements s (k). In view of the definitions (4.4.6) and (4.4.7), the set EZ ,µ ∪ RZ ,µ can be + + identified with the set of poles of the radial Green’s function G(ξ , ξ ′ ; k) and the poles of the generalized radial eigenfunctions e(x, k). Remark 4.31. The set R of resonances does not depend on the choice of the phase function which determines the Jost functions f± (k), the generalized radial eigenfunctions and the scattering matrix elements. 4.5. Formal “partial wave expansion” of the Green’s function For real E we know from the previous section that the spectrum of Kη = Kη (h, E) consists of an infinite number of simple eigenvalues µ1 (E) < µ2 (E) < µ3 (E) < . . . tending to infinity that extends to analytic functions of E in some neighborhood of the real line. We shall denote by φn,E the eigenfunctions Kη (E)φn,E (η) = µn (E)φn,E (η), normalized by ∫ 2 ∥φn,E ∥ = +π −π n ∈ N0 , |φn,E (η)|2 d η = 1 for E ∈ R+ and then extended analytically. We choose φn,E real for E real. Define K := F (ξ , η)HG . (4.5.1) Instead of solving (HG − E)u = f in L2 (M , F (ξ , η)d ξ d η) for E ∈ C \ σ(HG ), we look at the solutions of (K − F (ξ , η)E)u(ξ , η) = F (ξ , η) f (ξ , η). (4.5.2) We already know (see (4.1.2)) that (K − F (ξ , η)E)u(ξ , η) = KE u(ξ , η) = (Kξ + Kη )u(ξ , η). ¦ © Now, using the completeness of the orthonormal base φn,E possesses the expansion u(ξ , η) = ∑ n∈N0 n∈N0 for E ∈ R, u un (ξ , η) with un (ξ , η) := φn,E (η)ψn,E (ξ ), (4.5.3) 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 70 where ∫ ψn,E (ξ ) = +π −π φn,E (η)u(ξ , η) d η. This expansion extends to complex values of E by analyticity (note that no complex conjugate is involved, since φn,E is chosen real for E ∈ R). Analogously we get F (ξ , η) f (ξ , η) = ∑ φn,E (η)gn,E (ξ ), (4.5.4) n∈N0 where ∫ gn,E (ξ ) := +π −π φn,E (η)(F f )(ξ , η) d η. Substituting (4.5.3) and (4.5.4) into (4.5.2) one gets Kξ + Kη ∑ n∈N0 or equivalently ∑ φn,E (η) ∑ un (ξ , η) = φn,E (η) gn,E (ξ ) n∈N0 Kξ (E) + µn (E) ψn,E (ξ ) − gn,E (ξ ) = 0. (4.5.5) n∈N0 Remark 4.32. (4.5.5) extends to complex points E ̸∈ σ(H ), where Kξ (E) + µn (E) possesses an inverse Rn (E) similarly as (B.13) by means of the Green’s function defined in (4.4.7). By (4.5.5): ψn,E (ξ ) = Rn (E) gn,E (ξ ) ∫ ∫ = Gn (ξ , ξ˜; E) R+ (4.5.6) +π −π φn,E (e η)(F f )(ξ˜, ηe) d ηe d ξ˜. Combining (4.5.6) and (4.5.3) we obtain u(ξ , η) = ∑ ∫∫ φn,E (η) n∈N0 M0 Gn (ξ , ξ˜; E)φn,E (e η)(F f )(ξ˜, ηe) d ξ˜ d ηe and we read off the partial wave expansion for the Green function G(ξ , η; ξ˜, ηe; E) = ∑ φn,E (η)φn,E (e η)Gn (ξ , ξ˜; E)(cosh2 ξ˜ − cos2 ηe). (4.5.7) n∈N0 It would be of great interest to be able to prove that the sum converges in the sense of distributions in the product space D ′ (M0 ) ⊗ D ′ (M0 ). Then we could use our results on the analytic continuation of the Gn and of the angular eigenfunctions to give a meromorphic continuation of the G(ξ , η; ξ˜, η̃; E) in E to the second Riemann sheet (or k ∈ C− ). We believe that the lack of this proof is only a matter of time and cannot be included in the thesis just for the imminence of the bureaucratic deadlines. 4.6. RESONANCES FOR THE 2-DIMENSIONAL PROBLEM 71 Nevertheless, for each fixed N ∈ N, we can consider the restriction KN of the operator K to the subspace ΥN (E) := N ⊕ Φn (E) ⊗ L2 (R+ , cosh2 (ξ )d ξ ) n=0 (4.5.8) ⊂ L2 ([−π, π], d η) ⊗ L2 (R+ , cosh2 (ξ )d ξ ) where Φn (E) is the subspace spanned by φn,E . The relative Green’s function GN (ξ , η; ξ˜, ηe; E) = N ∑ φn,E (η)φn,E (e η)Gn (ξ , ξ˜; E)(cosh2 ξ˜ − cos2 ηe) n=0 is the truncated sum obtained from (4.5.7) and as a finite sum of well defined term is convergent. Moreover it follows from the results of the previous sections that it possesses a meromorphic continuation in E to the second Riemann sheet. 4.6. Resonances for the 2-dimensional problem The operator Kη defined by (4.2.4) has discrete spectrum µn (k 2 ) admitting an analytic continuation in k 2 := E in some neighborhood of the real axis. At the same time for each µ, the resolvent of the operator Kξ (µ, Z+ ) (see Remark 4.32) can be extended in terms of k to the negative complex plane, having there a discrete set of poles k m (µ). With the definitions given in Section 4.4 we set En := k ∈ C+ \ {0} | f+ (k, µn (k 2 )) = 0, e −ξ /2 e iϕ(ξ ,k,µn (k 2 )) ∈ L2 (R+ , cosh2 (ξ )d ξ ) . (4.6.1) If k ∈ En , we call k an eigenvalue of the quadratic eigenvalue problem for K(Z− , Z+ ). All other zeros of the Jost function f+ (k, µn (k)) are called resonances of K(Z− , Z+ ) and we denote them by ¦ © Rn := k ∈ D+ (α, β) \ En | f+ (k, µn (k 2 )) = 0 . (4.6.2) Proposition 4.33. The sets En and Rn are made by an at most countable number of elements k m ∈ D+ (α, β) ( m ∈ I ⊆ N ) of finite multiplicity such that f+ (k m , µn (k m )) = 0. Proof. f+ (k) and µn (k 2 ) being analytic functions of k , the statement is clear. □ Remark 4.34. Notice that if k 2 is an eigenvalue of the full operator K (or its restriction KN ), then it must be an eigenvalue of Kξ (Z+ , µn ) for some µn (k 2 ) (i.e. an element of En ). Remark 4.35. By definition it is obvious that En ∩ Rn = ;. Furthermore, it is clear looking at the asymptotic behavior of the solutions that it is impossible that k ∈ En and k ∈ Rn ′ for n ̸= n ′ . 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 72 Relying on the previous discussion and on Remark 4.29 we can switch from the k plane to the k plane and and refer to 2 EN := N ∪ En , RN := n=0 N ∪ Rn (4.6.3) n=0 as the sets of eigenvalues and resonances of KN . Moreover, in view of Remark 4.30, the points of EN ∪ RN contained in D+ (α, β) ∩ D− (α, β) are in one-to-one correspondence with the poles of the scattering matrix elements sn (k) := s (k, µn ) and with the poles of the Green’s functions Gn (ξ , ξe; k) := G(ξ , ξe; k, µn (k)) for n ∈ {0, . . . , N }. Remark 4.36. If we suppose that (4.5.7) is convergent, we can refer to E := ∞ ∪ En , R := n=0 ∞ ∪ Rn (4.6.4) n=0 as the sets of eigenvalues and resonances of K . As for the restricted operator, in view of Remark 4.30, the points of E ∪R contained in D+ (α, β)∩D− (α, β) are in one-toone correspondence with the poles of the scattering matrix elements sn (k) := s (k, µn ) and with the poles of the Green’s functions Gn (ξ , ξe; k) := G(ξ , ξe; k, µn (k)). 4.7. Computation of the resonances of Kξ Consider the equation 0 = Kξ (h)ϕ(ξ ) = −h 2 ∂ξ2 ϕ(ξ ) − Z+ cosh(ξ )ϕ(ξ ) − E cosh2 (ξ )ϕ(ξ ) (4.7.1) with the condition ϕ′ (0) = 0. The potential V (ξ ; Z+ , E) := −Z+ cosh(ξ ) − E cosh2 (ξ ) has Taylor expansion around ξ = 0 given by E 2 e ξ + e −ξ − e ξ + e −ξ 2 4 E 2 Z+ =− 2 + ξ 2 + O (ξ 4 ) − 2 + 2 + (2ξ )2 + O (ξ 4 ) 2 4 Z+ = −Z+ − E − E + ξ 2 + O (ξ 4 ) 2 V (ξ ; Z+ , E) = − Z+ = A − ω 2 ξ 2 + O (ξ 4 ), q Z where A := −Z+ − E and ω = E + 2+ . Z Let now E + 2+ > 0. We would like to apply the theory [BCD87a, BCD87b, BCD88] and [Sjö87] to get the resonances from the eigenvalues en (h) = h(2n + 1)ω (n ∈ N) 4.7. COMPUTATION OF THE RESONANCES OF Kξ 73 of the harmonic oscillator Hos c = −h 2 ∂ξ2 + ω 2 ξ 2 , according to An (h, E, Z+ ) = −Z+ − E − i h(2n + 1)ω + O (h 3/2 ). Remark 4.37. In the above discussion there is a major bug: in [BCD87a, BCD87b, BCD88] and [Sjö87] it is essential to assume that the potential is bounded, and this is clearly false in (4.7.1). The problem stressed by the previous remark can be solved. With the change of variable given by y := sinh(ξ ) : R+ → R+ we change the measure from cosh2 (ξ ) d ξ Æ to y 2 + 1 d y but at the same time the differential equation of Kξ (Z+ , µ) takes the form −h 2 (y 2 +1)∂y2 u(y)−h 2 y∂y u(y)+(µ−k 2 (y 2 +1)−Z+ Æ y 2 + 1)u(y) = 0. Note that µ will correspond to an eigenvalue of the angular equation Kη , and as such it will be an analytic function of E . Moreover it will be real for real values of E (see Section 4.2). With the ansatz u(y) := Æ 4 1 y2 + 1 v(y) we can rewrite the differential equation in Liouville normal form as y2 + 1 2 2 −h ∂y v(y) + V (k, Z+ , µ, h; y)v(y) = 0 Æ 4 y2 + 1 (4.7.2) where V (k, Z+ , µ, h; y) := −k 2 − Æ Z+ 2 y +1 + µ 1 + y2 − y2 − 2 4(y 2 + 1)2 h 2. This potential V has the following properties: • • • • • it is smooth in R+ ; it is bounded; it is analytic in a cone centered at the positive real axis; it has a non-degenerate global maximum at y = 0; around the maximum V can be expanded in Taylor series as V (k, Z+ , µ, h; y) = A − ω 2 y 2 + O (y 4 ), where A := −Z+ − k + µ − 2 h2 2 and ω = q µ + 45 h 2 − Z+ . 2 Therefore it satisfies the assumptions of [BCD87a, BCD87b, BCD88] and [Sjö87] and we can use their theories to approximate the resonances through the eigenvalues of the harmonic oscillator according to An (h, E, Z+ , µ) = −Z+ − k 2 + µ − i h(2n + 1)ω + O (h 3/2 ). (4.7.3) 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 74 This given, we have a zero of (4.7.2) if v is identically 0 or if An = 0, this last condition permits us to identify with some approximation the resonances of our operator Kξ (Z+ , µ). From this first formula one can have a first approximation of the resonances En = kn2 in orders of ℜ(µ) ≫ 0 as follows ℑEn ℜEn p = (2n + 1)h ℜµ + ℑµ + O (ℜµ)−1/2 q = ℜµ − Z+ + O (ℜµ)−1/2 . (4.7.4) Remark 4.38. The approximation (4.7.3) identifies the resonances generated by the top of the potential (at ξ = 0) and these corresponds to the resonances generated by the classical closed hyperbolic trajectory bouncing between the two centers (see Remark 1.5). Remark 4.39. In Section 2.6 we have seen that for Z+ < 0, |Z+ | < Z− , there is for small energies a region of the phase-space characterized by closed orbits related to a local minimum of the potential. We expect in this case the appearance of some shape-resonances at exponentially small distance in h from the real axis (see [HS80, HCS84] and [HS96, Chapter 20]). We plan to study the existence and the distribution of these other resonances in a future work. 4.8. Eigenvalues asymptotics and resonant regions for Z− = 0 near the bottom of the spectrum As we did previously, before studying the general system, let us have a look to the simplest case Z− = 0. With a proper renaming of the constants and the notation of (4.2.2), in [MS54, Section 2.331] it is proved that Theorem 4.40. For δ → +∞ and n ∈ N ( p an (δ) = −2δ + (4n + 2) δ + O (1), p bn+1 (δ) = −2δ + (4n + 2) δ + O (1). Thus we have as a direct consequence the following theorem. Corollary 4.41. In the limit h ↘ 0 and for every E > 0 we have ( p µ+ (h, E, 0) = (2n + 1) E h + O (h 2 ), n p (h, E, 0) = (2n + 1) E h + O (h 2 ). µ− n+1 We can use this result in combination with (4.7.3). We obtain the equation for the resonance energies by setting −An (h, E, Z+ , µ+ (h, E, 0)) = 0. m Neglecting the error terms and writing E = k 2 we have s 2 k +Z+ −(2n+1)k h +i h(2m+1) (2n + 1)k h + 5h 2 4 − Z+ 2 = 0. (4.8.1) 4.9. RESONANCES FOR Z− > 0 75 4.9. Eigenvalues asymptotics and resonant regions for Z− > 0 near the bottom of the spectrum Notice that we can always define Z− in such a way that it is non-negative. In presence of the Z− term the equation Kη ψ(η) = 0 assumes the form 0 = −h 2 ∂η2 ψ(η) + −µ + Z− cos(η) + E cos2 (η) ψ(η), (4.9.1) with periodic boundary conditions on [−π, π]. Remark 4.42. We can use Corollary 4.41 and the Min-Max principle to compare the eigenvalues obtained for the Z− = 0 case with this more general one, but this is far from giving an accurate estimate. Lemma 4.43. In the limit h ↘ 0 and for E > 0 we have p p −Z− +(2n +1) E h +O (h 2 ) ≤ µn (h, E, Z− ) ≤ Z− +(2n +1) E h +O (h 2 ). Proof. It follows immediately after rewriting Kη ψ(η) = 0 in the Mathieu-like form: 0 = −∂η2 ψ(η) − 2µ − E 2h 2 ψ(η) + 2 E 4h 2 cos(2η)ψ(η) + Z− h2 cos(η)ψ(η). □ To obtain better estimates for the spectrum in orders of small h we use the ε-quasimodes [AK99, Laz93]. If A is a self-adjoint operator on D(A) in a Hilbert space H , then for ε > 0 one can call a pair e Ee ∈ D(A) × R, ∥ψ∥ e = 1, ∥(E − E) e ≤ε e ψ∥ ψ, an ε-quasimode. Note that with this notation an eigenfunction ψ with eigenvalue E is a 0-quasimode. e Ee implies that the distance between Ee The existence of an ε-quasimode ψ, and the spectrum of A fulfils dist σ(A), Ee ≤ ε. (4.9.2) In particular there exist an eigenvalue E of A in the interval [Ee − ϵ, Ee + ϵ] for ϵ ≤ ε, if in that interval we know that the spectrum is discrete. In our case we want to replace A with an operator of the form P h := −h 2 d2 d x2 + V (x) (4.9.3) 2 on an interval [−a, b ] ⊆ R (a, b > 0) and with V (x) := x4 + W (x) and W (x) := ∑∞ a x m ( m0 > 2) entire of order 1 and finite type. m=m m 0 Remark 4.44. The assumptions on V imply that it has a global non-degenerate minimum in 0. 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 76 It is a well known fact that for W = 0 P1 Dn1 = En1 Dn1 with En1 := n + 12 and Dn1 the normalized Hermite Polynomials Dn1 (x) := (−1)n x 2 d n − x 2 e4 e 2, p d xn n! 2π n ∈ N. (4.9.4) It is equally well known that P h Dnh = Enh Dnh (4.9.5) 1 1 with Enh := hEn1 and Dnh := h − 4 Dn1 h − 2 x . Lemma 4.45. (Dnh , Enh ) are O(h m0 /2 )-quasimodes for P h . Proof. By assumption V (x) = m0 ∥W · Dnh ∥ = O h 2 . x2 4 + W (x). We need to prove that Before entering in the details of the computation we want to remark that the space of entire functions of order 1 and of finite type forms an algebra and that we can always write Dn1 as Pn (x)e − ∫ ∥W · Dnh ∥2 = b −a x2 2 for an appropriate polynomial Pn (x) of degree n . 1 x 2 2 − 14 W (x) h Pn h − 2 x e − 2h d x ∫ ∞ 1 x2 ≤ 2h W 2 (x) Pn2 h − 2 x e − h d x ∫∞ 0 1 2 ≤2 W 2 h 2 y Pn2 (y)e −y d y . {z } | 0 − 12 We can rewrite W 2 (x) = ∑∞ =:I h b m x m with the appropriate b m . W 2 being and entire function, we can bound the coefficients b m using the following lemma m=2m0 [Lev96, Lemma 1, p. 5]: Lemma 4.46. If f (z) = max f (z) < e Ar ∑∞ c z m=0 m m is an entire function and for r big enough K |z|=r is fulfilled, then for m big enough |c m | < eAK m m K . In particular, for an entire function of order 1 and finite type σ , one has K = 1 and A = σ , and the inequality reads: For all m > M |c m | < σ e m m (4.9.6) 4.9. RESONANCES FOR Z− > 0 Moreover, for each k ∈ N one has ∫ ∞ 2 yk e −y 2 dy = Γ k +1 and thus ∫ I h = lim 2 M →∞ = lim 2 M →∞ = lim 2 M →∞ ∞ 0 , 2 0 77 M ∑ 2 h m b m y m Pn2 (y)e −y d y m=m0 M ∑ ∫ m ∞ C y h bm m+2n +O y m+2n−1 e −y 2 dy 0 m=m0 m + 2n + 1 m + 2n h m bm C Γ +O Γ 2 2 m=m0 M ∑ For m > M we can apply (4.9.6) and Stirling formula n n p δ Γ(n) = 2πn e 12n , 0 < δ < 1, e to obtain for m > M1 ≥ M σ e m m + 2n + 1 m + 2n + 1 < |C | Γ |C b m |Γ 2 m 2 !m m+1 p +n δ σe m 2 e 6(m+2n+1) = 2π |C | m e m m −(n+1) δ 1 2n + 1 2 e 2 = C1 σ m + e 6(m+2n+1) 2 2m m m m = C1 σ m γ12 γ22 −(n+1) δ e 6(m+2n+1) −→ 0. m→∞ Thus in particular the sum converges absolutely and we have Ih = h m0 ∞ ∑ bm h m−m0 m=m0 m + 2n + 1 C2 Γ = h m0 C3 , 2 and consequently m0 ∥W · Dnh ∥ = O h 2 . (4.9.7) □ Sadly our potential has in general two non-degenerate minima at the points ±η∗ := ± arccos − Z− 2E ∈ [−π/2, π/2] Z2 where the potential reaches the value − 4E− , thus we cannot directly apply the previous strategy as is (see Figure 4.1). Pushed by the intuition that the eigenvalues will be moved toward minV for h becoming small, we construct our quasimodes to be concentrated near one of the minima. Let the intervals ∆+ and ∆+ be two open o i ⊂ ∆+ and ∆+ neighborhoods of the rightmost minima such that ∆+ is contained in o o i 78 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) Figure 4.1. Shape of Z− cos(η) + E cos2 (η) in [−π, π]. the positive axis and is strictly separated from 0. Fix χ+ ∈ C0∞ (R) such that χ+ = 1 in ∆+ and χ+ = 0 in R \ ∆+ . o i (x−x )2 Lemma 4.47. Let P h be as in (4.9.3) but with V (x) := 4 ∗ + W (x) and ∑ a (x − x∗ ) m ( m0 > 2) entire of order 1 and finite type. Define W (x) := ∞ m=m0 m 1 1 ψnh (x) := h − 4 Dn h − 2 (x − x∗ ) χ+ (x) = Dnh (x − x∗ )χ+ (x), where χ+ is the characteristic function defined in the previous paragraph. Then (ψnh (x), Enh ) is an O h 3/2 -quasimode for P h . Proof. Applying the operator to ψnh we have ′′ P h ψnh = − h 2 ψnh + (x − x∗ )2 ψnh + W ψnh 4 ! 2 (x − x ) ′′ ∗ = −h 2 Dnh + Dnh + W Dnh χ+ 4 1 ′ − h 2 2h − 2 Dnh χ+′ + Dnh χ ′′ 1 (4.9.3) h h ′ = En ψn + W ψnh − h 2 2h − 2 Dnh χ+′ + Dnh χ ′′ . For what concerns W ψnh we are in the situation of Lemma 4.45, thus we can use equation (4.9.7) as bound to obtain m0 ∥W · ψnh ∥ = O h 2 . We need now to take care of the last error term. For this last term the inequality c 2 − h2 h ′′ − 12 h ′ ′ ≤ hc e χ + D χ h 2h D 1 n n + holds with proper c1 , c2 > 0 (that depend only on n and χ+ ). Thus this term integrated on [a, b ] will give an error that can be bounded with any polynomial order of decay, in particular we can choose it to be m0 1 ′ 2 h 2h − 2 Dnh χ+′ + Dnh χ ′′ = O h 2 . □ 4.9. RESONANCES FOR Z− > 0 79 We need now to transform our equation into something like V (x) in the previous theorem. We already know the two minima ±η∗ . If we expand V (η) := Z− cos(η) + E cos2 (η) in the neighborhood of those minima we obtain V (x) = − Z−2 4E +E 1− Z−2 4E ! 2 (η ± η∗ )2 + W (η ± η∗ ) (4.9.8) for a suitable entire W with m0 = 3 and of order 1 and finite type. To simplify a bit the notation let us call A := − Z−2 4E , v u 2 u Z − B := tE 1 − 2 . 4E We focus for the moment only the localization near the rightmost minima, i.e. we choose (η−η∗ ). With the unitary transformation Z defined by change of variable z(η) := p 2B(η − η∗ ), the eigenvalue equation (4.9.1) is transformed into 0 = K z ψ(z) := 2B e+ −h 2 ∂η2 ψ(z) + µ z2 4 ! ! f(z) ψ(z) , +W (4.9.9) where e= µ 1 2B (−µ + A) f is entire with m = 3 and of order 1 and finite type. If in the spirit of the and W 0 previous lemmas we define ψenh (z) := Dnh (z)χ (z), enh µ := A + 2B n + 1 2 h, where χ (z) is the transformed of the cut-off localized in the neighborhood of η∗ , enh ) is an O(h 3/2 )-quasimode for K z and thus if then the couple (ψenh , µ ψnh (η) := Z −1 ψenh Z (η), h enh ) defines an O(h 3/2 )-quasimode for Kη . the couple (ψn± ,µ Exactly the same happens if we look near the other minimum, i.e. if we choose the case (η + η∗ ). In other words in the limit of h ↘ 0 the spectrum of Kη consists enh in the limit. of pairs µ− (h), µ+ (h) with the same asymptotics µ n n We have proved the following. Z Theorem 4.48. Let E > 2− > 0. Define v u 2 Z Z−2 u − enh := − + tE 1 − 2 (2n + 1) h. µ 4E 4E There exists an eigenvalue µnh of Kη and a constant c such that h h e = c h 3/2 . µ − µ n n (4.9.10) 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 80 enh − 2c h 3/2 , µ enh + 2c h 3/2 Moreover, the interval µ ues of Kη . contains at least two eigenval- Remark 4.49. It can be proved by standard methods involving the IMS formula [CFKS87, Chapter 3.1] and Agmon estimates [Agm82] that the distance between the eigenvalues in each pair is of the order exp(−C /h) with C ∈ R+ . We can use this result in combination with (4.7.3). Neglecting the error terms, we obtain the equation for the resonance energies by setting An (h, E, Z+ , µ+ (h, E)) = 0. m (4.9.11) They are given by the solutions of s Z−2 Z−2 (2m + 1) h + E− 4E vs u u Z−2 Z−2 Z+ t (2m + 1) h − + i h(2n + 1) E− − = 0. 4E 4E 2 −E − Z+ − 4E (4.9.12) Remark 4.50. For Z− = 0 we recover (4.8.1) of the previous section. In Section (4.8) on the other hand the approximation error is of order O(h 2 ) instead of O(h 3/2 ). Z For 0 < E < 2− the bottom of the potential is reached at π and thus we expand the potential around this other point. It turns out that in this case the eigenvalues are approximated by È bnh := E − Z− + µ Z− 2 − E (2n + 1) h. (4.9.13) And the resonances in this range can be computed accordingly from the equation b+ An (h, E, Z+ , µ (h, E)) = 0. m Remark 4.51. This approach gives good results if we stay localized near the bottom of the potential: in this case we can find an approximation for the eigenvalue up to an order of any integer power of h . The deficiency of this approach lies in the fact that we have no control on the relative error between n and h . We need therefore to find a different approximation scheme that keeps track of the mutual relation between the parameters. 4.10. High energy estimates We consider the potential in the form V (x) = E cos2 (x) + Z− cos(x). Substituting this value in the formulae given in Theorem D.4 we have ∫ π −π ∫ V (x)d x = Eπ and π −π V 2 (x)d x = 3E 2 π 4 + πZ−2 4.11. NUMERICAL INVESTIGATIONS 81 and thus the eigenvalues µ2m+1 and µ2m+2 can be represented as p µ = (m + 1)h + E + Z−2 − E2 ! 1 4(m + 1)h 4 16(m + 1)3 h 3 (4.10.1) 1 1 +O +o . m5 h 5 m3 h Therefore we can estimate µ2m+1 and µ2m+2 with ! 2 E E 1 µ = (m + 1)2 h 2 + + Z−2 + 2 4 8(m + 1)2 h 2 (4.10.2) 1 1 +O +o . m4 h 4 m2 We can thus approximate the resonances En,2m+1 and En,2m+2 solving An (h, E, Z+ , µ2m+1 (h, E)) = 0. (4.10.3) More explicitly, for fixed n and up to errors of orders 3 h2, (m h)−4 and m −2 , we get the expression − E 2 2 2 2 − Z+ + (m + 1) h + + i (2n + 1)h v u u t Z−2 + E4 8(m + 1)2 h 2 (m + 1)2 h 2 + E − Z+ 2 2 + Z−2 + E4 8(m + 1)2 h 2 = 0. Remark 4.52. We cannot hide the term (m + 1)2 h 2 in the error term of order h 3/2 because we want to analyze the asymptotic behavior for m ≥ C /h (C ∈ R+ ) and that term is rather big compared with h . 4.11. Numerical investigations In the previous sections we have explicitly written three implicit equations to approximate the value of the resonances in terms of the atomic numbers n and m (and of course of the parameters h , Z+ and Z− ). Modern symbolic algebra software like Mathematica, Maple and SAGE are often able to find explicit solutions of implicit equations. These are generally incredibly long and complex but easily computable and plottable. In this section we want to investigate the qualitative structure of the resonances using the approximations given by (4.9.11) and (4.10.3). In view of Remark 4.38 and Remark 4.39 we know that at least for certain values of the charges Zi we are not describing all the resonances of the system, on the other hand the additional resonances should appear only for small ℜ(E). Therefore we want to consider ℜ(E) big enough to be sure that we are considering an energy region in which all the resonances are generated by the classical closed hyperbolic trajectory between the centers. 82 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) In this case equation (4.7.4) implies that ℜ(µ m ) must be big and thus it is evident from (4.9.10), (4.9.13) and (4.10.2) that m must be big. The quasi-mode approximation obtained in Section 4.8 and 4.9 are valid only for small values of m and h , therefore they are automatically excluded from the analysis. In some sense we are lucky because it was not possible to solve Equation (4.9.11) (algebraically or numerically) for small h , arbitrary Z+ and Z− ̸= 0. In Figure 4.2(a) and 4.2(b) we plotted all the approximated resonances obtained from (4.9.11) setting Z− = 0. We plotted all the values including the one in not allowed regions of energies or where we have no control on the error. In these picture we can see a particularly interesting behavior. In particular for big values of m we recover the structure shown by the resonances approximated with (4.10.3): see Figure 4.3(a) and Figure 4.3(b). The physically interesting resonances are the one nearer to the real axis, this because they can be measured in experiments. Thus to keep ℑ(E) as small as possible we will consider small values of n (see (4.7.4)). Remark 4.53. Unless differently specified, in the plots we consider n = 0, 1, 2, 3 and m ∈ {⌈C /h⌉ + k | k = 0, 1, 2, . . . , 20}. The values of Z+ , Z− , h and C will be specified in the title or in the caption of the plots. For practical reasons we will plot the resonances in the plane (ℜ(E), −ℑ(E)). The software used for the simulations is Wolfram Mathematica 8. Equation (4.10.3) has two couples of solutions (S+ , L+ ) and (S− , L− ), specular w.r.t. the real axis. They correspond respectively to the resonances and the anti-resonances, i.e. the resonances defined inverting the roles of the incoming and outgoing waves v± in the construction of Section 4.4. We restrict our analysis to the resonances (S+ , L+ ). The two sets S+ , L+ ∈ C− characterize two different energy regions, this meaning that the resonances in S+ have relatively small real part if compared to the resonances in L+ (see Figure 4.3(a) and Figure 4.3(b)). The structure that we find is extremely regular and the first question that arises is if we are really computing the resonances associated with energy values on the critical line L+2 discussed in Section 2.3. For each computed resonance En,m we can use the approximation obtained in (4.10.2) to estimate the associated constant of motion Kn,m . We can thus superimpose the points (ℜ(E), ℜ(K)) to the bifurcation diagram and visualize how they are related. As shown in Figure 4.4, the energy parameters appear to lay exactly upon L+2 , giving a strong hint on the correctness of the result. Another reasonable question at this point regards the order of growth of the resonances in n and m . For a system that is classically hyperbolic we can borrow an ansatz from physics. In scattering physics the study of quantum resonances is generally performed with the Gutzwiller trace formula. One of the standard results is that the imaginary part of the energy is proportional to the sum of the Lyapunov exponents of the closed hyperbolic trajectories [GAB95]. It is well known that the Lyapunov exponent of the bounded orbits of a Coulombic two-centers system diverges 4.11. NUMERICAL INVESTIGATIONS 83 2 1 -2 2 4 6 8 -1 -2 (a) Case Z+ = 2. 0.6 0.4 0.2 5 10 15 - 0.2 - 0.4 - 0.6 (b) Case Z+ = −2. Figure 4.2. Solutions of (4.9.11) with Z− = 0, h = 0.01, n = 0, . . . , 4, m = 1, . . . , 250. 10 84 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) (Z+ , Z− ) = (2, 4) (Z+ , Z− ) = (−2, 4) (Z+ , Z− ) = (4, 2) (Z+ , Z− ) = (−4, 2) (a) Resonances in L+ . (Z+ , Z− ) = (2, 4) (Z+ , Z− ) = (−2, 4) (Z+ , Z− ) = (4, 2) (Z+ , Z− ) = (−4, 2) (b) Resonances in S+ . Figure 4.3. Resonances for h = 0.05 and C = 10. 4.11. NUMERICAL INVESTIGATIONS p 85 like ℓ(E) = E ln(E) (see [KK92, Proposition 5.6]). In this sense it is reasonable to normalize the real and imaginary part of the resonances in L+ (or S+ ) dividing them by ℓ(ℜ(E)). In this way it is possible to investigate, at least qualitatively the formally expected behavior. The numerics confirms the expected behavior. It is evident from Figure 4.5(a) and 4.5(b) that the renormalized resonances look like distributed on a regular lattice of points with (almost perfectly) aligned and equispaced real and imaginary parts. Notice moreover that the vertical spacing of the imaginary parts is d = O(h) and the distance between the real axis and the resonances with smaller imaginary part is approximately d /2, as expected from the harmonic oscillator perturbation used to approximate the resonances. 4. THE TWO-CENTERS SYSTEM ON L2 (R2 ) 86 H Z + = 2 , Z - = 4L H Z + = - 2 , Z - = 4L 1.4 2.0 1.2 1.0 1.5 0.8 1.0 0.6 0.4 0.5 0.2 0.0 0.0 60 65 70 75 75 0 80 85 90 95 0 - 20 - 20 - 40 - 40 - 60 - 60 æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò - 80 æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò - 80 60 65 70 æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò -100 75 75 (a) (Z+ , Z− ) = (2, 4), C = 4, m = 400, . . . , 430. 80 85 90 95 (b) (Z+ , Z− ) = (−2, 4), C = 7, m = 700, . . . , 730. H Z + = 4 , Z - = 2L H Z + = - 4 , Z - = 2L 1.4 2.5 1.2 2.0 1.0 0.8 1.5 0.6 1.0 0.4 0.5 0.2 0.0 0.0 65 70 75 80 85 85 90 95 100 105 0 0 - 20 - 20 - 40 - 40 - 60 - 60 - 80 - 80 ò ì ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì à æ ò ì æ à ò ì à æ ò ì æ à ò ì à æ ò ì æ à ò ì æ à ò ì à æ ò ì à æ ò ì æ à ò ì æ à ò ì à æ ò ì à æ ò ì æ à ò ì æ à ò ì æ à ò ì æ à ò ì à æ ò æ à 65 70 75 80 (c) (Z+ , Z− ) = (4, 2), C = 4, m = 400, . . . , 430. æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò -100 85 85 90 95 100 æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò æ à ì ò 105 (d) (Z+ , Z− ) = (−4, 2), C = 7, m = 700, . . . , 730. Figure 4.4. Comparison of the resonances in L+ for h = 0.001 (plot above) and their projection on the bifurcation diagram (plot below). 4.11. NUMERICAL INVESTIGATIONS 87 H Z + ,Z - L =H 2 ,4L ò ò ò ò ò H Z + ,Z - L =H - 2 ,4L ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ì ì ì ì ì ì ì ì ì ì ì à à à à à à à à à à à æ æ æ æ æ æ æ æ æ æ æ 0.0035 0.004 0.0030 ì ì ì ì ì ì ì ì ì ì ì 0.0025 0.003 0.0020 0.0015 à à à à à à à à à à à æ æ æ æ æ æ æ æ æ æ æ 0.002 0.0010 0.001 0.0005 2.331 0.0035 ò ò 2.332 ò 2.333 2.334 H Z + ,Z - L =H 4 ,2L ò ò ò ò ò ò ò 2.269 2.270 2.271 H Z + ,Z - L =H - 4 ,2L 2.272 ò ò ò ò ò ò ò ò ò ò ò ì ì ì ì ì ì ì ì ì ì ì à à à à à à à à à à à æ æ æ æ æ æ æ æ æ æ æ ò 0.005 0.0030 0.0025 2.268 2.335 ì ì ì ì ì ì ì ì ì ì ì à à à à à à à à à à à 0.004 0.0020 0.003 0.0015 0.002 0.0010 0.0005 æ æ æ 2.359 æ æ 2.360 æ æ æ 2.361 æ æ 2.362 æ 0.001 2.363 2.231 2.232 2.233 2.234 2.235 (a) Plot for for E ∈ L+ . H Z + ,Z - L =H 2 ,4L ò ò ò ò ò ò H Z + ,Z - L =H - 2 ,4L ò ò ò ò ò 0.006 0.005 ì ì ì ì ì ì ì ì ì ì ì à à à à à à à à à à à æ æ æ ò æ æ 1.4470 ò ò ò æ æ æ æ 1.4475 ò ò ò ò ò ò ò 0.005 ì ì ì ì ì ì ì ì ì ì ì 0.003 à à à à à à à à à à à ò ò æ 0.001 æ æ æ æ æ æ æ æ æ æ æ 1.4480 H Z + ,Z - L =H 4 ,2L ò æ ò ò ò 1.5865 ò 1.5870 1.5875 H Z + ,Z - L =H - 4 ,2L ò ò ò ò ò ò ò ò ò ò ò ì ì ì ì ì ì ì ì ì ì ì à à à à à à à à à à à æ æ æ æ æ æ æ æ æ æ æ 0.007 0.007 0.006 0.006 ì ì ì ì ì ì ì ì ì ì ì à à à à à à à à à à à æ æ æ æ æ æ æ æ æ æ æ 0.005 0.005 0.004 0.004 0.003 0.002 0.002 0.001 ò 0.002 1.4465 0.003 ò 0.004 0.002 0.001 ò 0.006 0.004 0.003 ò 1.3828 1.3830 1.3832 1.3834 1.3836 1.3838 0.001 1.3840 1.6566 1.6568 1.6570 1.6572 (b) Plot for E ∈ S+ . Figure 4.5. Renormalized resonances E/ℓ(ℜ(E)) with parameters h = 0.001, C = 9 and m = 9000, . . . , 9010. 1.6574 1.6576 CHAPTER 5 Back to the three dimensional problem 5.1. Asymptotic analysis of the solutions of A lλ In view of the results of the preceding chapter, we rewrite the radial equation (3.6.1) of the three dimensional problem in normal form. For the moment we don’t worry about the domains of definition of the operator. With the definition u(ξ ) := p 1 sinh(ξ ) v(ξ ), (5.1.1) the differential equation A lλ u = 0 takes the following form −h 2 v ′′ (ξ ) + µ + h2 4 + h 2 λ2 sinh2 (ξ ) − k 2 cosh2 (ξ ) − Z+ cosh(ξ ) v(ξ ) = 0. (5.1.2) Clearly, for λ = 0 equation (5.1.2) has the same form of (4.3.1) and all the asymptotic results of Section 4.3 can be applied. The radial incoming and outgoing waves of (3.6.1) can be thus defined for k ∈ C± by u± (ξ , k) := p 1 sinh(ξ ) v± (ξ , k) (5.1.3) where v± (ξ , k) is given by Theorem 4.11 with properly redefined constants. For u± (ξ , k) we have up to a constant factor the asymptotic expansion u± (ξ , k) = e −ξ e ±iϕ(ξ ,k/h) (1 + o(1)) as ξ → ∞. (5.1.4) The existence of an analytic continuation in k through the positive real axis follows then by Theorem 4.20. Remark 5.1. The first description of (5.1.4) for the general two-center problem is given in [Lea86] and [MS54]. The solutions u± are explicitly represented of as slowly convergent sums of special functions. In this way the authors had a deep control on the regularity of u± for complex ξ and could use the known asymptotic form of the considered special functions to estimate the asymptotic behavior of the u± . With these methods, on the other hand, the regularity of u± with respect to k , µ and Z+ was inaccessible. Remark 5.2. The asymptotic expansions of [Lea86] confirms that the asymptotic (5.1.4) continues to hold in the case λ ̸= 0, but we cannot apply Theorem 4.20 or extend its proof in a straightforward way because of the singularity of h 2 λ2 sinh2 (ξ ) at the origin. We shall thus find a way to regularize the equation or transform it in a more manageable one. 89 90 5. BACK TO THE THREE DIMENSIONAL PROBLEM We proceed like in (4.7.2) and apply the change of variables induced by y = sinh(ξ ). Then (5.1.2) transforms to e − h 2 y∂y v(y) e −h 2 (y 2 + 1)∂y2 v(y) + µ+ h2 4 + h 2 λ2 y2 2 2 − k (y + 1) − Z+ Æ ! e = 0, y + 1 v(y) 2 and with the ansatz e =Æ v(y) 4 we get 1 y2 + 1 w(y) h2 2 2 2 µ + y +1 2 k Z+ /h λ −h 2 Æ − 2 4 − 2 2 ∂y + 2 + Æ w(y) = 0. 4 h y2 + 1 y 2 + 1 y + 1 y (y + 1) 2 If we get rid of the term outside the brackets, we obtain the equation h2 2 2 2 µ + k Z /h λ + ∂y2 w(y) + 2 + Æ − 2 4 − 2 2 w(y) = 0. (5.1.5) 2 h y + 1 y (y + 1) y +1 (5.1.5) is particularly helpful because it fits in the framework considered by [AK92, Sections 3,4,5]. In particular in [AK92] it is proved that for k ∈ C± the incoming and outgoing waves are well defined functions, are uniquely determined by the asymptotic behavior k w± (y, k) = exp ±i y(1 + o(1)) (1 + o(1)) as y → ∞, h and admit an analytic extension in the energy parameter k through the real axis for arg ±k < π. If we apply the transformations needed to obtain (5.1.5) in the opposite direction we get that w± (y, k) are transformed to u± (ξ , k) having exactly the asymptotic form (5.1.4) and admitting an analytic extension in k through the positive real axis. Remark 5.3. The fact that for λ ̸= 0 there exists only one regular solution in 0 that is identified (up to a constant) by decaying at 0 like |y|λ and that is entire in k and µ is a standard consequence of Frobenius theory [Tes00, p. 105]. Additionally, in [AK92] are given explicit asymptotic bounds for the decay rates of u± and u0 in term of big values of λ. Now, for k ∈ C± the waves u± are in L2 (R+ , cosh2 (ξ ) sinh(ξ )d ξ ) and thus for λ ̸= 0 and any fixed µ it is possible to give a rigorous definition of the radial Green’s function, the scattering elements and the generalized eigenfunctions exactly as in Section 4.4. Moreover, if µn (λ, k) (n ∈ N0 ) are the eigenvalues of the angular operator, for each fixed λ and n we can define the sets En,λ and Rn,λ respectively of eigenvalues and of resonances of the radial Schrödinger operator Anλ defined by (3.5.3). These definition can be given exactly as in Section 4.6, with the care of using cosh2 (ξ ) sinh(ξ )d ξ instead of cosh2 (ξ )d ξ as measure of the L2 space. 5.2. ESTIMATE FOR THE EIGENVALUES IN THE CASE Z− = 0 91 Moreover for every λ and n the remarks and the propositions of Section 4.6 continue to hold and we can define Eλ := ∞ ∪ En,λ , Rλ := n=0 ∞ ∪ Rn,λ , (5.1.6) n=0 as the sets of eigenvalues and resonances of the Schrödinger operators Kλ defined in (3.2.1). We only have to prove the convergence of the partial wave expansion (3.7.6) for a fixed λ, in particular we can choose λ = 0. Then the convergence of the sum (3.7.6) would result estimating the decay of the Green’s function in λ using the decay at 0 of the regular solutions as in [AK92] (compare with Remark 5.3). This would give meaning to the second separation discussed in Section 3.2 and therefore we could define the sets of eigenvalues and resonances of the two-center Schrödinger operator H by means of E := ∪ Eλ and R := λ∈Z ∪ Rλ . (5.1.7) λ∈Z Remark 5.4. As for the planar case (see (4.5.8)), we can define a truncated operator in some subspace depending on a finite number of values of n and λ with convergent partial wave expansion (3.7.6). Its resonances belong to a subset of R and its eigenvalues to a subset of E . On the other hand, our approximations in this case are not as good as for the planar case. In fact the singularity appearing in equation (5.1.5) for λ ̸= 0 prevented us from giving a good numerical description of the resonances. Remark 5.5. Using the estimates developed in [AK92, Sections 3, 4 and 5] is it possible to define a truncated operator depending only on a finite number of values of n and with convergent partial wave expansion. As in the previous remark its resonances belong to a subset of R and its eigenvalues to a subset of E . Nevertheless, we expect that the structure of the resonances set for the three dimensional problem will be more involved than for the two dimensional one. In particular it is not possible to exclude the degeneracy of the resonances. A main intuition on this problem could be given by the description of the bifurcation diagrams for non planar motions. Sadly this was not developed in the thesis even if we hope to complete the puzzle adding the analysis of this last case in the next future. The following two sections present the estimates of the eigenvalues of the angular operator that we were able to find. 5.2. Estimate for the eigenvalues in the case Z− = 0 Remark 5.6. We want to approximate the eigenvalues and the eigenfunctions of A xλ y(x) = µ h2 y(x). 92 5. BACK TO THE THREE DIMENSIONAL PROBLEM With the following ansatz, coming from Frobenius theory of regular singular points [Tes00, p.105], y(x) = (1 − x 2 )λ/2 u(x), one finds for the u h λ −(1 − x 2 ) 2 (1 − x 2 )u ′′ (x) − 2x(λ + 1)u ′ (x) i µ E 2 Z− + − λ(λ + 1) − x − x u(x) = 0. h2 h2 h2 Let Z− = 0. If we set p γ := E z := , h p 2γ x, v(z) := u(x), L := we get the following equation for the v : 1− z2 ! z ′′ v (z) − 2(λ + 1) 2γ 2γ ′ v (z) + L − z2 4 h −2 µ − λ(λ + 1) 2γ , ! v(z) = 0. p Thus, formally setting 1/γ = h/ E = 0, we get a Quantum Harmonic Oscillator like equation, i.e. ′′ v (z) + L − z2 ! v(z) = 0. 4 Let us proceed now rigorously. Let n o R := y ∈ D(A xλ (h)) f (x) = (1 − x 2 )λ/2 u(x) with u entire . (5.2.1) The eigenvalue differential equation A xλ (h)y(x) = µλn (h) h2 y(x). for 0 ̸= y ∈ R then reads λ 2 −(1 − x ) (1 − x 2 )u ′′ (x) − 2x(λ + 1)u ′ (x) 2 + µλk h2 − λ(λ + 1) − E h2 x2 − Z− h2 ! (5.2.2) x u(x) = 0. For the moment we follow [MS54, Chapter 3.25] and study the symmetric problem, thus we set Z− = 0. We consider only λ ≥ 0. For λ < 0 is enough to replace all the appearences of λ with |λ|. Thinking to the previous remark, we set p y λp (x) := (1 − x 2 )λ/2 D p ( 2γ x) and M pλ (γ ) := (2 p + 1)γ + λ(λ + 1) (5.2.3) 5.2. ESTIMATE FOR THE EIGENVALUES IN THE CASE Z− = 0 93 where p γ := γ (E, h) := E h , and for p ∈ N D p (x) := (−1) p e x2 4 dp dx p x2 e− 2 (5.2.4) are the Parabolyc Cylinder Functions. Remark 5.7. Notice that h → 0 implies γ → ∞ Remark 5.8. The ′ appearing in what follows means that we are taking the derivative with respect to the explicitly written variable. Using the fact that D ′′p (x) + p+ 1 2 − x2 4 ! D p (x) = 0, we have A xλ (h) y λp + M pλ (h) y λp = f p where λ f pλ = − 1 − x 2 2 z 2 D p ′′ (z) + 2z(λ + 1)D p ′ (z) , z= p 2γ x. Theorem 5.9. ∥ f pλ ∥ ∥y λp ∥ = O (1) (γ → ∞). Proof. First of all we have that p z 2 D p ′′ (z) = x 2 D p ′′ ( 2γ x) and p z D p ′ (z) = xD p ′ ( 2γ x), thus p λ 2 ′′ p ′ 2 2 x D p ( 2γ x) + 2x(λ + 1)D p ( 2γ x) ∥ f pλ ∥ − 1 − x = . λ p ∥y λp ∥ 1 − x 2 2 D ( 2γ x) p By means of the binomial formula we can perform the expansion 1 − !2 λ !λ−n λ λ ∑ z z2 z2 1 − =1− +O p = 2γ 2γ γ2 2γ n=0 n 94 5. BACK TO THE THREE DIMENSIONAL PROBLEM Writing x = pz , we have 2γ 1 ∥ f pλ ∥2 = p 2γ 1 =p 2γ 1 =p 2γ C1 =p 2γ ∫ p2γ p 1 − !2 λ 2 z 2 ′′ ′ p z z D p (z) + 2(λ + 1)D p (z) d z 2γ − 2γ ∫ p2γ 2 1 2 ′′ ′ 1 + O z z D (z) + 2(λ + 1)D (z) dz p p p γ − 2γ ! ∫ +∞ 2 1 z 2 z D ′′p (z) + 2(λ + 1)D ′p (z) d z + O γ 3/2 −∞ ! 1 +O . γ 3/2 where C1 is a finite positive constant and we used the fact that the tails are given by a bounded integral that decays exponentially in γ and that we are computing the integral of a positive even function. In the same way we have 1 ∥y λp ∥2 = p 2γ ∫ p2γ p 1 − − 2γ z p 2γ !2 λ ∥D p ∥2 +O D p (z) d z = p 2γ 1 γ 3/2 ! , where ∥D p ∥2 is finite, strictly positive and exactly computable. Thus, when γ → ∞ ∥ f pλ ∥ ∥y λp ∥ p → C1 ∥D p ∥ < ∞. □ We need the following theorem. Theorem 5.10. Let R be a complex linear space with an Hermitian positivedefined scalar product. Let F be a linear operator defined on D(F ) ⊂ R dense. Let its discrete spectrum is given by a countable set of simple eigenvalues λn = P2 (n) + O (1), where n ∈ N0 or n ∈ Z and P2 a polynomial of degree exactly 2 s.t. for n1 ̸= n2 it is P2 (n1 ) ̸= P2 (n2 ). The corresponding eigenvalues build an orthonormal system in R 〈yi , yk∗ 〉 = δi k , and there exist two constants α, β (+∞ > α > β > 0) such that for every function f in the space β2 ∑ n |〈 f , yn∗ 〉|2 ≤ 〈 f , f 〉 ≤ α2 For z ∈ D(F ) and arbitrary λ F z + λz = f ∑ n |〈 f , yn∗ 〉|2 . 5.3. ESTIMATE FOR THE EIGENVALUES IN THE CASE Z− ̸= 0 95 with f ∈ R . Moreover if λ = λ m is an eigenvalue of F , then ∗ 〉=0 〈 f , ym and ∥z∥ ≤ α 1 β minλn ̸=λ |λn − λ| ∥ f ∥. Proof. This theorem is a particular case of [MS54, Chapter 1.5, Satz 1], we refer the reader to the reference for the proof. □ p It follows from Theorem 5.10 that, in the limit γ = µλn (E, h) λ = O (1). min M p (E, h) − n h2 E h → +∞, we have Thus in particular there exist a constant C such that p = n −λ+C. Theorem 5.11. Thus there exist C such that for n ≥ λ and for every E > 0, lim µλn (E, h) h h↘0 = p E(2(n − λ + C ) + 1) Proof. It follows directly from the fact that µλn (Z− , E, h) h2 = (2(n − λ + C ) + 1) γ (E, h) + λ(λ − 1) + O (1). □ Remark 5.12. In principle, one expects the eigenvalues to be quadratic in n , what happens in fact is that in the semi-classical limit only the Harmonic Oscillator like part of the potential becomes relevant, hiding this relation. In fact for E2 ≪ 1, h see [MS54, Chapter 3.24], in the limit n → ∞ one finds µλn (Z− , E, h) h2 = n(n + 1) − 1E 2 h2 +O 1 n2 , where one recalls the quadratic dependence in n of the eigenvalues and its independence from the h dependent factor. 5.3. Estimate for the eigenvalues in the case Z− ̸= 0 Like in the previous subsection, we set p y λp (x) := (1 − x 2 )λ/2 F pδ ( 2γ x) where p γ := γ (E, h) := E h , and M pλ (γ ) := (2 p + 1)γ + λ(λ + 1), (5.3.1) δ := δ(Z− , E, h) := Z− 2E r γ Z− = Æ p , 2 2 2hE E 96 5. BACK TO THE THREE DIMENSIONAL PROBLEM and F pδ (x) := D p+[δ 2 ] (x + 2δ), D p (x) := (−1) p e x2 4 dp dx p x2 e− 2 . (5.3.2) Remark 5.13. Notice that δ = 0 corresponds to Z− = 0. Using the fact that F pδ ′′ (x) + p+ 1 2 − x2 4 ! − δ x − {δ 2 } F pδ (x) = 0, where {·} ∈ [0, 1) is the fractional part function, we have A xλ (h) y λp + M pλ (h) y λp = f p where for z = p 2γ x ′ λ ′′ f pλ = − 1 − x 2 2 z 2 F pδ (z) + 2z(λ + 1) F pδ (z) − {δ 2 }F pδ (z) . Theorem 5.14. For E > Z− /2 ∥ f pλ ∥ ∥y λp ∥ = O (1) (γ → ∞). Proof. First of all we have that ′′ p ′′ z 2 F pδ (z) = x 2 F pδ ( 2γ x) and ′ p ′ z F pδ (z) = x F pδ ( 2γ x), thus ′ p p λ 2 δ ′′ p δ 2 δ 2 2 x Fp ( 2γ x) + 2x(λ + 1) F p ( 2γ x) − {δ }F p ( 2γ x) ∥ f pλ ∥ − 1 − x . = λ p ∥y λp ∥ 1 − x 2 2 F δ ( 2γ x) p By means of the binomial formula we can expand 1 − z − 2δ p 2γ !2 λ !n !λ−n λ λ ∑ Z−2 Z− z2 1− 2 + z − = n E 2γ 4E n=0 ! Z−2 zZ− 1 = 1+ − 2 +O , E γ 4E 5.3. ESTIMATE FOR THE EIGENVALUES IN THE CASE Z− ̸= 0 97 p , we have and writing x = z−2δ 2γ 1 ∥ f pλ ∥2 = p 2γ 1 =p 2γ 1 =p 2γ ∫ p2γ +2δ 1 − p − 2γ +2δ ∫ p2γ +2δ − z − 2δ p 2γ p 2γ +2δ 1− Z−2 4E 2 ∫∞ 1+ !∫ zZ− ∞ −∞ E − !2 λ 2 Ξ (z, p, δ, λ) d z Z−2 4E 2 +O ! 1 γ Ξ2 (z, p, δ, λ) d z Ξ2 (z, p, δ, λ) d z 1 Z− +p zΞ2 (z, p, δ, λ) d z + O 2γ E −∞ ! ! Z−2 C2 1 = 1− 2 p +O 4E 2γ γ 3/2 ! 1 γ 3/2 where C2 is a finite positive constant and Ξ(z, p, δ, λ) := z 2 D ′′ p+[δ 2 ] (z) + 2z(λ + 1)D ′ p+[δ 2 ] (z) − {δ 2 }D p+[δ 2 ] (z). To perform this computation we used the following facts: (1) p lim (± 2γ + 2δ) = lim γ →+∞ p γ →∞ 2γ ±1 + Z− 2E = ±∞ iff E > Z− 2 so that the integral over the tails is exponentially decaying in γ ; (2) Ξ2 (z, p, δ, λ) is even, thus zΞ2 (z, p, δ, λ) integrated on a symmetric interval gives a vanishing term; (3) C2 is non negative because it is the integral of a positive even function on a symmetric interval. In the same way we have !2 λ z − 2δ 2 p D p+[δ 2 ] (z) d z 2γ − 2γ +2δ ! ! ∫∞ Z−2 1 1 zZ− (z) d z + O =p 1+ − 2 D2 p+[δ 2 ] E 4E 2γ −∞ γ 3/2 ! Z−2 ∫ ∞ 1 D2 (z) d z =p 1− 2 p+[δ 2 ] 4E 2γ −∞ ! ∫ 1 Z− ∞ 1 2 +p (z) d z + O zD p+[δ 2 ] 2γ E −∞ γ 3/2 2 ! ! Z−2 D p+[δ 2 ] 1 = 1− 2 +O . p 4E 2γ γ 3/2 1 ∥y λp ∥2 = p 2γ ∫ p2γ +2δ 1 − p 98 5. BACK TO THE THREE DIMENSIONAL PROBLEM Thus, in the limit γ → +∞ ( h ↘ 0), ∥ f pλ ∥ ∥y λp ∥ p → C2 ∥D p+[δ 2 ] ∥ . It follows from Theorem 5.10 that, in the limit γ = µλn (E, h) λ = O (1). min M p (E, h) − n h2 p E h □ → +∞, we have Therefore there exists a constant C such that p + δ2 = n − λ + C p = n − λ − δ2 + C . Theorem 5.15. There exists a constant C such that for n ≥ λ + δ 2 ≥ λ+ Z−2 p 8hE E and for every E > µλn (E, h) = − Z−2 4E + Z− , 2 or, equivalently, in the limit h ↘ 0 p E 2 n − λ − {δ 2 } + C + 1 h + O h 2 . Proof. It follows directly from the fact that δ 2 = δ 2 − {δ 2 } and µλn (Z− , E, h) h2 = 2(n − λ − δ 2 + C ) + 1 γ (E, h) + λ(λ − 1) + O (1). □ Remark 5.16. Sadly this is a really bad approximation, giving no hope to develop a numerical study of the resonances as in the previous chapter. CHAPTER 6 Final considerations Before discussing the open problems and the possible further directions of work, we want to give a summary of the main results of this thesis. In the classical picture, the novelty is given by Theorem 2.9 with the description of the bifurcation diagram for the planar motion for positive energies for arbitrary values of the charges. This is integrated in the subsequent section with a detailed analysis of the structure of the trajectories in configuration space. Most of the operator theoretical analysis of the separated equations of the quantum mechanical problem is new, nevertheless it is based on well established theories presented in the Appendices. The limit point criterion obtained in Theorem 3.11 should be new as far as we know. A new important results is given by Theorem 4.11 where it is proven that it is possible to define the concept, central in scattering theory, of incoming and outging waves for the radial separated equation. They are characterized by their asymptotic form using a WKB ansatz that involves a unique globally defined phase function. This result is strong in the sense that the approximation is not affected by the Stokes phenomenon near the turning points. Even more important is Theorem 4.20. In this theorem it is proved that the incoming and outgoing waves admits an analytic extension in the energy parameter to the second Riemann sheet. The major consequence of this theorem is the possibility to extend the Green’s function, the generalized eigenfunctions and the scattering matrix elements to the second Riemann sheet, making it possible to define the resonances for the two center system for the first time (see Sections 4.4 and 4.6). Moreover our construction integrates the one made in [AK92] and permits to characterize and define the resonances in the three dimensional case (see Chapter 5). Of great interest are even the results of Section 4.7, 4.9 and 4.10. Even if we have not a good mathematical control on them, they permit to localize numerically the resonances and give a qualitative description of their structure (see Section 4.11). Despite our efforts there are still many open question that we would try to address in the future. First of all in this work the proof of the convergence of the partial wave expansion of the Green’s function is missing for both the two and the three dimensional case. This is a important step to make the whole analysis completely rigorous. For the moment we only know that at least numerically it seems to converge. It is important to remark here that the missing point to prove the convergence of (3.7.6) is the convergence of the planar partial wave expansion (4.5.7). Secondly, as stressed in Remark 4.38 and 4.39, we have given a description of the resonances associated to the classical closed hyperbolic trajectory bouncing 99 100 6. FINAL CONSIDERATIONS between the centers. We expect that another class of resonances appear for Z+ < 0, |Z+ | < Z− , namely the shape resonances associated to energy parameter in the region I<a described in Section 2.6. Moving further from the two-centers model, a first interesting question arises from Remark 1.3. A diatomic molecule in Thomas Fermi approximation [LS77] looks like the two-centers operator (1.1.1) perturbed by a smooth potential W . In this case it is known that the closed hyperbolic trajectory between the centers is still present [KK92, Kna02] and the non-trapping condition fails to hold [CJK08]. It would be interesting to know if the resonances are still present and share the same structure as the resonances for the two-centers problem. In presence of the perturbation W the system is no more separable, thus we cannot apply the methods of this thesis. One possibility would be to proceed similarly as Gérard and Sjöstrand in [GS87]. Another big interesting question is the investigation of the resonances for the n centers problem for n ≥ 3. In this case is present a Cantor set of classical hyperbolic trajectories [KK92, Kna02] and the non-trapping condition fails to hold [CJK08]. Therefore one expect the resonances to be present and to be distributed in some complicated way. There are only few known examples presenting a similar structure that have been investigated rigorously (see [SZ07] and [NZ05]) and they suggests that the resonances are present and are distributed in a fractal set. In this case, anyhow, a huge effort is required to properly extend the techniques needed to deal with the problem. APPENDIX A A small introduction to the theory of Sturm-Liouville operators To proceed we need some theorems and preliminary definitions (cfr. [Wei80, pp. 247-257], [Tes09, pp. 84-85, 181-195]). Let I := (a, b ) ⊂ R define an interval (eventually unbounded), moreover set AC (a, b ) := f ∈ C (I ) f (x) = f (c) + ∫ x c g (t )d t , c ∈ I , g ∈ L1l oc (I ) the space of all the absolutely continuous functions. Consider a formal operator of the type τ f := 1 r {−( p f ′ )′ + q f } (A.1) where the coefficients p , q and r satisfy the following assumptions: (1) p , q and r are real-valued continuous functions defined on I ; (2) p is continuously differentiable; (3) p(x) > 0 and r (x) > 0 for all x ∈ I . The differential equation τ f = 0 is called Sturm-Liouville equation. If a is finite and if p −1 , q and r are in L1 ((a, c)) for some c ∈ I , then the Sturm-Liouville equation (A.1) is called regular in a , otherwise is called singular. Similarly for b . If it is regular (resp. singular) in both a and b , then it is called regular (resp. singular ). We call fundamental solutions of a Sturm-Liouville differential equation τ on [a, b ] the two solutions ϕ1 , ϕ2 that satisfy the following boundary conditions ϕ1 (0) = ϕ′2 (0) = 0 and ϕ′1 (0) = ϕ2 (0) = 1. (A.2) It is possible to define operators on L2 (I , r (x)d x) using differential equations such as τ . The maximal operator T induced by τ is defined by D(T ) = { f ∈ L2 (I , r (x)d x) | f , f ′ ∈ AC (a, b ), τ f ∈ L2 (I , r (x)d x)} T f =τf for f ∈ D(T ). The minimal operator T0 induced by τ is defined by the formula D(T0 ) = { f ∈ D(T ) | supp( f ) = K ⊂ I , K compact} T0 f = τ f for f ∈ D(T0 ). Remark A.1. When there is no risk of confusion, we will write L2 (I , r ) in place of L2 (I , r (x)d x) to shorten the notation. 101 102 A. A SMALL INTRODUCTION TO THE THEORY OF STURM-LIOUVILLE OPERATORS Theorem A.2. Let τ be as in (A.1). The operator T0 is symmetric and has equal deficiency indices, i.e., T0 has self-adjoint extensions. Theorem A.3. Let τ be as in (A.1). All self-adjoint extensions of T0 have the same essential spectrum. If T0 is semibounded, then all self-adjoint extensions of T0 are semibounded. Theorem A.4 (The Weyl alternative). Let τ be a Sturm-Liouville differential equation defined on I , and let c ∈ I . Either • every solution u of the equation (τ − z)u = 0 lies in L2 ((c, b ), r (x)d x) for every z ∈ C; or • for every z ∈ C there exists at least one solution u of the equation (τ − z)u = 0 for which u ∈ / L2 ((c, b ), r (x)d x). In the second case, for every z ∈ C \ R there exists (up to a factor) exactly one solution u of the equation (τ − z)u = 0 for which u ∈ L2 ((c, b ), r (x)d x). According to Weyl we say that in the first case we have the limit circle case (LCC) at b and in the second case we have the limit point case (LPC) at b . Similarly a corresponding theorem holds for the boundary point a from which the corresponding definitions of limit point case and limit circle case follow. For x ∈ I and for f , g ∈ C 1 (I , C), now, we define [ f , g ] x := p(x) f ′ (x)g (x) − f (x)g ′ (x) = −W ( f , g , x) (A.3) where W is the modified Wronskian. In the following theorem, borrowed from [Wei80] (see also [Tes09]), we write LPC for Limit Point Condition and LCC for Limit Circle Condition. Theorem A.5. Let τ be a Sturm-Liouville operator (A.1). Moreover, let λ ∈ R, and let v and w be non-vanishing real solutions of the equation (τ − λ)u = 0. (1) The operator Tv,w defined using the separate boundary conditions by D(Tv,w ) = { f ∈ D(T ) | [v, f ]a = 0 if LCC at a, [w, f ] b = 0 if LCC at b } Tv,w f = T f for f ∈ D(Tv,w ). is a self-adjoint extension of T0 . Note that if we have LPC at a and/or b , then the index v and/or w is meaningless. (2) For z ∈ C \ R the resolvent R z = (z − Tv,w )−1 is an Hilbert-Schmidt operator of the c -invariant form R z g (x) = 1 W (ua , u b ) ( ∫ u b (x) a + ua (x) x ua (y)g (y)r (y) d y ∫ b x ) u b (y)g (y)r (y) d y , where for some c ∈ (a, b ), ua and u b are the solutions of the equation (τ − z)u = 0 uniquely determined up to a factor by the conditions • [v, ua ]a = 0 if we have LCC at a , respectively ua ∈ L2 ((a, c), r ) if we have LPC at a , A. A SMALL INTRODUCTION TO THE THEORY OF STURM-LIOUVILLE OPERATORS 103 • [w, u b ] b = 0 if we have LCC at b , respectively u b ∈ L2 ((c, b ), r ) if we have LPC at b . (3) All eigenvalues of Tv,w are simple. (4) If we have the LCC at both a and b , then Tv,w has a pure discrete spectrum and the eigenfunctions form an orthonormal basis. Notice that if one defines the operator by means of coupled boundary conditions, the eventual discrete spectrum has at most multiplicity 2 but in general is not simple. In fact the following theorems can be proved. Theorem A.6. Let τ be a Sturm-Liouville operator (A.1) with p(x) > 0. Assume that the self-adjoint realization T is defined by means of the boundary conditions ¨ u(b ) = α u(a) p(b )u ′ (b ) = α1 p(a)u ′ (a) with α > 0. By u j we denote the eigenfunction corresponding to the eigenvalue λ j (for doubledegenerate eigenvalues the choice of the corresponding eigenfunctions is arbitrary). • u0 has no zero in [a, b ], u2n+1 and u2n+2 both have exactly 2(n +1) zeros in [a, b ). • We have λ0 < λ1 ≤ λ2 < . . . ≤ λ2n < λ2n+1 ≤ λ2n+2 < . . . . Proof. The proof can be found in [Wei87, Theorem 13.7, p. 206] and [Eas75, Theorem 2.3.1 and Theorem 3.1.2]. □ Remark A.7. The periodic boundary conditions are the special case α = 1. As in the separated case, the eigenvalues of Sturm-Liouville problems can be compared to obtain bounds on the spectrum. Theorem A.8. Let r (x) = 1. Let λ1,n (n ≥ 0) denote eigenvalues in the periodic problem over [a, b ] when p(x) and q(x) are replaced by p1 (x) and q1 (x), where p1 (x) ≥ p(x) and q1 (x) ≥ q(x). Then we have λ1,n ≥ λn for all n . Proof. Given in a more general form in [Eas75, Theorem 2.2.2, p. 23] □ For even potentials we can characterize the periodic boundary conditions in terms of separated boundary conditions. In fact the following theorem holds. Theorem A.9. Consider the differential equation y ′′ + Q(x)y = 0 (A.4) where Q(x) is an even (real or complex valued) function of the real variable x , piecewise continuous in every finite interval and with minimal period π. Let ϕ1 (x) and ϕ2 (x) be its fundamental solutions as defined by (A.2). Then there exists a nontrivial periodic solution of (A.4) which is 104 A. A SMALL INTRODUCTION TO THE THEORY OF STURM-LIOUVILLE OPERATORS (1) (2) (3) (4) even and of period π, if and only if ϕ′2 (π/2) = 0. odd and of period π, if and only if ϕ1 (π/2) = 0. even and of period 2π, if and only if ϕ2 (π/2) = 0. odd and of period 2π, if and only if ϕ′1 (π/2) = 0. Proof. The proof can be found in [MW79, Theorem 1.1] □ APPENDIX B Singular second-order Sturm-Liouville operators with complex coefficients The Weyl LPC-LCC classification presented in the previous Appendix can be generalized to Sturm-Liouville problems of the form (A.1) on an interval [a, b ), a < b ≤ ∞, where (1) r > 0, p ̸= 0 a.e. on [a, b ) and r, 1/ p ∈ L1loc ([a, b )); (2) p, q are complex valued, q ∈ L1loc ([a, b )) and ¨ Q := co q(x) r (x) « + ρ p(x) x ∈ [a, b ), ρ ∈ (0, ∞) ̸= C, (B.1) where co denotes the closed convex hull. Similarly as in the real case, the end point b is called singular if at least one of b = ∞ or ∫ b r (x) + a 1 | p(x)| + |q(x)| d x = ∞ holds. The general theory and the proofs of the theorems of this section can be found in [BMEP99] unless a different source is explicitly specified. We will present here only the results that we need and adapt them to our notation. We will continue to denote the Sturm-Liouville differential equation with τ as in (A.1). The complement in C of the closed convex set Q has one or two connected components. For λ0 ∈ C \ Q , denote by K = K(λ0 ) its (unique) nearest point in Q and denote by L = L(λ0 ) the tangent to Q at K if it exists or otherwise any line touching Q at K . Then if the complex plane is subjected to a translation z → z − K and a rotation through an appropriate angle η = η(λ0 ) ∈ (−π, π], the image of L coincides with the imaginary axis and the images of λ0 and Q lie in the new negative (w.r.t. the real line) and non-negative half-planes respectively: i.e. for all x ∈ [a, b ) and r ∈ R+ , ℜ ρ p(x) + q(x) r (x) −K e iη ≥ 0 and ℜ (λ0 − K) e i η < 0. (B.2) For such admissible η, K (corresponding to some λ0 ∈ C \ Q ), we define the region ¦ © Λη,K := λ ∈ C | ℜ (λ0 − K) e i η < 0 . (B.3) Then C \ Q is the union of the half-planes Λη,K over the set S of admissible η, K . 105 106 B. SINGULAR COMPLEX STURM-LIOUVILLE PROBLEMS Let θ, ϕ be the solutions of −( p y ′ )′ + q y = λr y satisfying ¨ (B.4) ϕ(a, λ) = sin(α), θ(a, λ) = cos(α), ′ pϕ (a, λ) = − cos(α), pθ′ (a, λ) = sin(α), (B.5) where α ∈ C. In fact we will consider only the case α = 0 or α = π/2, so we suppose from now α ∈ {0, π/2} without specifying it all the time unless it is necessary. Remark B.1. The values α = 0, π/2 correspond respectively to the usual Dirichlet and Neumann problems. For different values of α’s the set of admissible η, K and the region Q need to be slightly redefined and the situation can be partially different from the one depicted below, the general theory in detail can be found in [BMEP99] and the references therein. It is well known that the linearly independent solutions ϕ and θ are at least continuous in (x, λ) and for every fixed x are entire functions of λ with their derivatives [CL55]. Moreover every solution ψ of (B.4) except for ϕ is, up to a constant multiple, of the form ψ(x, λ) = θ(x, λ) + m(λ)ϕ(x, λ) (B.6) for some function m depending only of λ in some complex region. Remark B.2. As in the real case one can define LPC and LCC conditions in terms of the number of solutions of (B.4) lying in L2 ([a, b ), r (x)d x). As in the real case, in the LPC the function m is uniquely defined while in the LCC there is a continuous number of possible m lying on some complex circle C b (λ) and being specified uniquely by the boundary conditions at b . We want to understand the analytical properties of such an m and its domain of definition. Before doing this we need to state how the Weyl alternative translates in this more general setting. Theorem B.3. For λ ∈ Λη,K , (η, K) ∈ S the Weyl circles converge either to a limit-point m(λ) or a limit-circle C b (λ). The following distinct cases are possible, the first two being sub-cases of the limit-point case. Case I: there exists a unique solution y of (B.4) satisfying ∫ b a iη ′ 2 2 ℜ e ( p|y | + (q − K r )|y| ∫ dx + b |y|2 r d x < ∞, (B.7) a and this is the only solution satisfying y ∈ L2 ([a, b ), r (x)d x). Case II: there exists a unique solution y of (B.4) satisfying (B.7) but all solutions of (B.4) are in L2 ([a, b ), r (x)d x). Case III: all solutions of (B.4) satisfy (B.7) and, hence, are in L2 ([a, b ), r (x)d x). Moreover this classification is independent of λ in the sense that (1) if all solutions of (B.4) satisfy (B.7) for some λ′ ∈ Λη,K , then all solutions of (B.4) satisfy (B.7) for all λ ∈ C; B. SINGULAR COMPLEX STURM-LIOUVILLE PROBLEMS 107 (2) if all solutions of (B.4) are in L2 ([a, b ), r (x)d x) for some λ′ ∈ C, then all solutions of (B.4) for all λ ∈ C. Remark B.4. In cases I I and I I I the classification appears to depend on the chosen K, η even assuming more restrictions on the coefficients of (B.4). We are ready to state the results on the regularity properties of the m function. Lemma B.5. In cases I and II, mη,K is analytic throughout Λη,K . In case I, the function defined by m(λ) = mη,K (λ) (λ ∈ Λη,K ) is well defined on each of the possible two connected components of C \ Q ; the restriction to a connected component is analytic on that set. In case I I I , given m0 ∈ C b (λ0 ), λ0 ∈ Λη,K , there exists a function mη,K that is analytic in Λη,K and mη,K (λ0 ) = m0 ; moreover, a function mη,K can be found such that mη,K (λ) ∈ C b (λ) for all λ ∈ Λη,K . Remark B.6. If α = 0 (or π), m(·) possesses an analogue of the Nevanlinna representation [AG93, Chapter 2.69, Theorem 2], like for the Titchmarsh-Weyl function of the symmetric case. That is not the case if α = π/2. Lemma B.7. Let λ, λ′ ∈ Λη,K and ψ(·, λ) = θ(·, λ) + m(λ)ϕ(·, λ), (B.8) where m(λ) is either the limit-point or an arbitrary point on D b (λ) in the LCC. Then lim [ψ(·, λ), ψ(·, λ′ )](β) = 0, (B.9) β→b with the modified Wronskian [, ] from (A.3). In case I, (B.9) continues to hold for all λ, λ′ ∈ C \ Q . Theorem B.8. For all λ, λ′ ∈ Λη,K ∫b ′ (λ − λ) ψ(x, λ)ψ(x, λ′ )r (x)d x = m(λ) − m(λ′ ); (B.10) a this holds for all λ, λ′ ∈ C \ Q in case I. Moreover, in cases II and III, for a fixed λ′ ∈ Λη,K , the function m(λ) can be extended as a meromorphic function in C with pole at λ if and only if ′ ∫ 1 + (λ − λ ) b ϕ(x, λ)ψ(x, λ′ )w(x)d x = 0. a Theorem B.9. Suppose that (B.4) is in case I. Define « ¨ q(x) Qc := co + ρ p(x) x ∈ [c, b ), ρ ∈ (0, ∞) , r (x) Q b := ∩ Qc . c∈(a,b ) (B.11) Then m(λ) is defined throughout C\Q , and has a meromorphic extension to C\Q b , with poles only in Q \ Q b . 108 B. SINGULAR COMPLEX STURM-LIOUVILLE PROBLEMS By means of the theory just presented it is possible to define the Green function and the resolvent associated to (B.4). For λ ∈ Λη,K , (η, K) ∈ S , define ( G(x, y; λ) = −ϕ(x, λ)ψ(y, λ), a < x < y < b , −ψ(x, λ)ϕ(y, λ), a < y < x < b , (B.12) where ϕ, ψ are the functions defined in (B.5) and (B.8). For f ∈ L2 ([a, b ), r (x)d x) we define ∫ Rλ f (x) := b G(x, y; λ) f (y)r (y) d y. (B.13) a Lemma B.10. Let λ ∈ Λη,K , (η, K) ∈ S . For Rλ defined in (B.13) the following properties are satisfied. (1) For x ∈ (a, b ), [ϕ, ψ](x) = [ϕ, ψ](a) = 1. (2) For f ∈ L2 ([a, b ), r (x)d x), p(Rλ f )′ ∈ ACloc ([a, b )) and a.e. x ∈ (a, b ) (τ − λ)Rλ f (x) = f (x), where τ is the differential equation (A.1) with coefficients fulfilling the conditions described at the beginning of this section. (3) For any λ′ ∈ C and for f ∈ L2 ([a, b ), r (x)d x) [Rλ f , ϕ(·, λ′ )](a) = 0. (B.14) Moreover, if λ′ ∈ Λη,K [Rλ f , ψ(·, λ′ )](b ) = 0. (B.15) Remark B.11. In general m , and hence ψ, depends on (η, K). However Lemma B.5 shows that in Case I the definition properly extends to C \ Q . Remark B.12. In case I, (B.15) holds for all λ, λ′ ∈ C \ Q . Theorem B.13. Let f ∈ L2 ([a, b ), r (x)d x) and λ ∈ Λη,K , (η, K) ∈ S . Then, in every case ∫ b ℜ e i η ( p|Φ′ |2 + (q − K r )|Φ|2 ) d x a ∫ ∫ b 1 b 2 iη 2 + ℜ((K − λ)e ) − ε |Φ| r d x ≤ |f | rdx 4ε a a for any ε > 0 where Φ := Rλ f and δ = dist(λ, ∂ Λη,K ). In particular, Rλ is bounded and ∥Rλ f ∥ ≤ 1 δ ∥f ∥ (B.16) where ∥ · ∥ is to be intended as the L2 ([a, b ), r (x)d x) norm. Moreover in cases II and III, Rλ is Hilbert-Schmidt for any λ ∈ Λη,K , (η, K) ∈ S . B. SINGULAR COMPLEX STURM-LIOUVILLE PROBLEMS 109 For some (η, K) ∈ S fix a λ′ ∈ Λη,K and set n D (e τ ) := u ∈ L2 ([a, b ), r (x)d x) | u, p u ′ ∈ ACloc ([a, b )), τu ∈ L2 ([a, b ), r (x)d x), o [u, ϕ(·, λ′ )](a) = 0 = [u, ψ(·, λ′ )](b ) , τeu := τu, u ∈ D (e τ) . (B.17) In general D(e τ ) will depend on λ′ , but in one case as clarified by the following theorem. Theorem B.14. Let n D1 := u ∈ L2 ([a, b ), r (x)d x) | u, p u ′ ∈ ACloc ([a, b )), τu ∈ L2 ([a, b ), r (x)d x), o cos(α)u(a) + sin(α) p(a)u ′ (a) = 0 . Then in Case I one has the equality D1 = D(e τ ), while in cases II and III, D1 is given ′ by the direct sum D1 = D(e τ ) + span(ϕ(·, λ )). Let now J be the conjugation operator u 7→ u . An operator T is called J symmetric if J T J ⊂ T ∗ and J -self-adjoint if J T J = T ∗ . Moreover T is m -accretive if ℜλ < 0 implies that λ ∈ ρ(T ), the resolvent set of T , and ∥(T −λ1l)−1 ∥ ≤ 1/|ℜλ|. If, for some K ∈ C and η ∈ (−π, π), e i η (T − K) is m -accretive, we shall say that T is quasi- m -accretive. Let σ(e τ ) denote the spectrum of τe, then the essential spectrum σe (e τ ) is defined as the complement in C of the set ∆(e τ ) := {λ ∈ C | (e τ − λ1l) is a Fredholm operator and ind(e τ − λ1l) = 0} , where an operator A is called Fredholm if its kernel and cokernel are finite-dimensional and its range is closed. The following theorem will give a meaning to the operator defined in (B.17). Theorem B.15. The operators defined in (B.17) for any λ′ ∈ Λη,K , (η, K) ∈ S are J -self-adjoint and quasi- m -accretive, and σ(e τ ) ⊆ C \ Λη,K . For any λ ∈ Λη,K m −1 (e τ − λ1l) = Rλ . In case I, σ(e τ ) ⊆ Q and σe (e τ ) ⊆ Q b . In Q \Q b , σ(e τ ) consists only of eigenvalues of finite geometric multiplicity, these points being the poles of the meromorphic extension of m defined in Theorem B.9. In cases II and II, Rλ is compact for any λ ∈ ρ(e τ ) and σ(e τ ) consists only of isolated eigenvalues (in C \ Λη,K ) having finite algebraic multiplicity. Remark B.16. Notice that in case of a real differential equation, everything coincides with the classical theory of Sturm-Liouville Operators with a singular endpoint. 110 B. SINGULAR COMPLEX STURM-LIOUVILLE PROBLEMS Despite its length, the following is an extremely useful result to understand in which case one equation belongs is the following theorem on asymptotic solutions of differential equations of the form (A.1). Theorem B.17 (Liouville-Green Theorem). Consider the Sturm-Liouville differential equation (B.4). Let p and Q := q − λr be nowhere zero complex valued functions and with locally absolutely continuous first derivatives in [x0 , ∞) for some x0 in their domain of definition. Assume the following conditions are fulfilled p • ( pQ)′ / pQ = o( Q/ p) as x → ∞, ′ • p −1/2 Q −3/2 ( pQ)′ ∈ L1 (x0 , ∞), • p −3/2 Q −5/2 ( pQ)′2 ∈ L1 (x0 , ∞), p • ℜ( Q/ p) have one sign in [x0 , ∞). Then (B.4) has solutions y1 and y2 such that up to a constant factor ∫ p x y1 = ( pQ)−1/4 exp x Q/ p d t as x → ∞, 0 ∫ p x as x → ∞, p y1′ = ( pQ)1/4 exp x Q/ p d t 0 p with similar formulae for y2 containing − Q/ p in the exponential term. Proof. See [Eas89, Corollary 2.2.1, p. 58] (B.18) (B.19) □ APPENDIX C Perturbation theory in a nutshell This chapter is mainly based on [RS78, Chapter XII.2] unless otherwise stated. A more general version of the theory can be found in [Kat95, Chapter VII]. A simplified reformulation particularly useful for applications to Mathieu and spheroidal operators can be found in [MS54, Chapters 1.4–1.6]. Let A and B be two operators densely defined on an Hilbert space H with domains D(A) and D(B) respectively. B is said to be A-bounded if (1) D(B) ⊃ D(A) (2) For some a and b in R and all ϕ ∈ D(A), ∥Bϕ∥ ≤ a∥Aϕ∥ + b ∥ϕ∥ The infimum of such a is called the relative bound of B with respect to A. If the relative bound is zero, B is said to be infinitesimally small with respect to A (B ≪ A). Theorem C.1 (Kato-Rellich Theorem). Suppose that A is self-adjoint, B is symmetric, and B is A bounded with relative bound a < 1. Then A+B is self-adjoint on D(A). Further, if A is bounded from below, then A + B is bounded from below. Let A be a self-adjoint operator. An operator C with D(A) ⊂ D(C ) is called relatively compact with respect to A if and only if C (A + i )−1 is compact. For this kind of operator one has the following property. Theorem C.2. Let A be a self-adjoint operator and let C be a relatively compact perturbation of A. Then (1) B = A + C defined on D(B) = D(A) is a closed operator; (2) if C is symmetric, B is self-adjoint; (3) σe s s (A) = σe s s (B). Let R ⊆ C be a connected domain and let T (β) be a closed operator with nonempty resolvent set ρ(T (β)) for each β ∈ R. • We say that T (β) is an analytic family if and only if for every β0 ∈ R, there is a γ0 ∈ ρ(T (β0 )) such that γ0 ∈ ρ(T (β)) for β near β0 and (T (β) − γ0 )−1 is an analytic operator-valued function of β near β0 . • We say that T (β) is an analytic family of type (A) if and only if (1) the operator domain D(T (β)) of T (β) is some set D independent of β; (2) for each ϕ ∈ D , T (β)ϕ is a vector-valued analytic function of β. Remark C.3. The number γ0 in the above definition plays no special role. 111 112 C. PERTURBATION THEORY IN A NUTSHELL One of the main properties of the analytic families of operators is clarified by the following theorem. Theorem C.4 (Kato-Rellich Theorem). Let T (β) be an analytic family. Let λ0 be a non-degenerate discrete eigenvalue of T (β0 ). Then, for β near β0 : • there is exactly one point λ(β) of σ(T (β)) near λ0 and this point is isolated and non-degenerate; • λ(β) is an analytic function of β and has a corresponding analytic eigenvector ϕ(β). Moreover, if T (β) is self-adjoint for β − β0 real, then ϕ(β) can be chosen to be normalized for β − β0 real. The following theorem gives a simple criterion for a family to be type(A). Theorem C.5. Let H0 be a closed operator with nonempty resolvent set. Define H0 + βV on D(H0 ) ∩ D(V ). Then H0 + βV is an analytic family of type (A) if and only if V is H0 -bounded. The main advantage of having an analytic family of closed operators, in particular of a family of type (A), is that it guarantees the existence of an analytic extension of the eigenvalues. In particular the following theorem holds. Theorem C.6. Let H0 + βV be an analytic family of type (A) in a region R. Then if 0 ∈ R and E0 is an isolated non-degenerate eigenvalue of H0 , then there is a unique point E(β) of σ(H0 + βV ) near E0 when |β| is small which is an isolated non-degenerate eigenvalue. Moreover E(β) is analytic near β = 0. The convergence radius can be estimated by means of the following theorem. Theorem C.7. Suppose that ∥V ϕ∥ ≤ a∥H0 ϕ∥ + b ∥ϕ∥. Let H0 be selfadjoint with an unperturbed isolated, non-degenerate eigenvalue λ0 . Define 1 ε := ε(λ0 ) := dist(λ0 , σ(H0 ) \ {λ0 }) 2 (C.1) r (a, b , λ0 , ε) = [a + ε−1 [b + a(|λ0 | + ε)]]−1 . (C.2) and Then the eigenvalue λ(β) of H0 + βV near λ0 is analytic at least in the circle of radius r (a, b , λ0 , ε). A huge simplification of the general theory can be achieved in presence of a self-adjoint analytic family of operator. Let T (β) be an analytic family of operators for β in a domain R ⊆ C symmetric with respect to the real axis. Suppose that (1) T (β) densely defined for all β ∈ R, (2) T (β)∗ = T (β). Then we call T (β) a self-adjoint analytic family. Remark C.8. It is clear from this definition that for all β∩R, T (β) is a closed self-adjoint operator. C. PERTURBATION THEORY IN A NUTSHELL 113 If T (β) is a self-adjoint analytic family on a domain R containing 0, any isolated eigenvalue λ of T (0) with finite multiplicity splits in one or more eigenvalues of T (β) that are analytic at β = 0. Every λ(β) of these analytic functions, with its eigenfunction, can be analytically extended along the real axis representing an eigenvalue of T (β) with its eigenfunction and this stays true even when the graph of λ(β) crosses the graph of another such eigenvalue, as long as the eigenvalue is isolated and has finite multiplicity. This feature is even stronger when the family is of type (A), in fact the following theorem holds [Kat95, Theorem 3.9, Chapter VII.3.5]. Theorem C.9 (Rellich Theorem). Let T (β) be a self-adjoint analytic family of type (A) defined for β in a neighborhood of an interval I0 of the real axis. Furthermore, let T (β) have compact resolvent. Then all eigenvalues of T (β) can be represented by functions which are analytic in I0 . More precisely, there is a sequence of scalar-valued functions λn (β) and a sequence of vector-valued functions ϕn (β), all analytic on I0 , such that for β ∈ I0 , the λn (β) represent all the repeated eigenvalues of T (β) and the ϕn (β) form a complete orthonormal family of the associated eigenvectors of T (β). Remark C.10. ([Kat95, Remark 3.10, Chapter VII.3.5]) Each λn (β) is analytic in a complex neighborhood of I0 but this neighborhood will depend on n , so that it is in general impossible to find a complex neighborhood in which all the λn (β) exists. Remark C.11. The statements concerning analyticity are false for two-parameter perturbations. A standard counterexample is given by the following family of linear operators in R2 α β T (α, β) := . β −α APPENDIX D Generalized Prüfer transformation in the semi-classical limit The method for establishing estimates is based on a modification of the Prüfer variables described in [Eas75, Chapter 4.1]. Consider a Sturm-Liouville differential equation on [x1 , x2 ] of the form (C (x)y ′ (x))′ + D(x)y(x) = 0 (D.1) in which C (x) and D(x) are real-valued, not necessarily periodic, differentiable and with piecewise continuous derivatives. Suppose also that C (x) and D(x) are positive p and define R(x) := C (x)D(x). If y is a non-trivial real-valued solution of (D.1), we can write R(x)y(x) = ρ(x) sin(θ(x)), where C (x)y ′ (x) = ρ(x) cos(θ(x)), (D.2) Æ R2 (x)y 2 (x) + C 2 (x)y ′2 (x), R(x)y(x) θ(x) := arctan . C (x)y ′ (x) ρ(x) := Up to now θ(x) is defined as a continuous function of x only up to a multiple of 2π, to solve this problem we select a point a0 ∈ [x1 , x2 ] and we stipulate that −π ≤ θ(a0 ) < π. Moreover, if y(a0 ) ≥ 0, we have by (D.1) that 0 ≤ θ(a0 ) < π. (D.3) Lemma D.1. Let a1 ∈ (a0 , x2 ] and let the other definitions be the one given above. Then ′ θ (x) = D(x) C (x) 1/2 + 1 (C (x)D(x))′ 4 C (x)D(x) sin(2θ(x)). (D.4) If y(x) has N zeroes in (a0 , a1 ] and y(a0 ) ≥ 0, then N π ≤ θ(a1 ) < (N + 1)π. (D.5) Proof. The theorem is proved in [Eas75, Chapter 4.1]. □ We want to apply (D.2) to equation (4.9.1). In particular we still keep ourselves a bit more general saying that we want to apply the transform to h 2 ( p(x)y ′ (x))′ + (µ − V1 (x))y(x) = 0 115 (D.6) 116 D. GENERALIZED PRÜFER TRANSFORMATION IN THE SEMI-CLASSICAL LIMIT where V1 (x) is not necessarily the potential V (x) that we are analyzing but it does have the same period 2π. Since we are concerned with the limit µ → ∞ (parametrically depending on h ), we can consider µ large enough to have µ − V1 (x) > 0 in [−π, π]. In the new case (D.6) the two functions θ(x) and ρ(x) will depend on µ and h as well as x . If we write θ(x) as θ h (x, µ), (D.4) becomes θ′h (x, µ) = 1 s µ − V1 (x) h + p(x) 1 µ p ′ (x) − ( p(x)V1 (x))′ 4 (µ − V1 (x)) p(x) (D.7) sin(2θ h (x, µ)). A first consequence of (D.7) is that as µ → ∞ v u u 1 − Ve (x) µ t 1 θ′h (x, µ) = + O(1), h p(x) 1 2 (D.8) where Ve1 (x) := V1 (x)/µ. Moreover, if y(x) has period 2π we have θ h (π, µ) − θ h (−π, µ) = 2kπ (D.9) for an integer k . Lemma D.2. Let f (x) be integrable over [−π, π]. Let c be a constant. Let θ h (x, µ) satisfy (D.7). Then ∫π f (x) sin c θ h (x, µ) d x −→ 0 −π as µ → ∞ (and/or h↘0). The same results holds with sin c θ h (x, µ) replaced by cos c θ h (x, µ) . Proof. To keep the equations compact let’s drop the µ dependence of θ h (x, µ) in the rest of the proof. Fix any ε > 0. Let g (x) be a continuously differentiable function such that ∫ π −π Then | f (x) − g (x)| d x < ε. ∫ ∫ π π g (x) sin c θ h (x) d x . f (x) sin c θ h (x) d x < ε + −π −π Define v u u G(x) := g (x)t p(x) 1 − Ve1 (x) . (D.10) D. GENERALIZED PRÜFER TRANSFORMATION IN THE SEMI-CLASSICAL LIMIT 117 Then by (D.8) h ′ g (x) sin c θ h (x) d x = 1 G(x) sin c θ h (x) θ h (x)d x + O 1 −π µ 2 −π µ2 h ′ π G (x) cos c θ h (x) −π = 1 c µ2 ∫π h ′ − G (x) cos c θ h (x) d x + O 1 . −π µ2 ∫ π Hence ∫ h π ∫ π h g (x) sin c θ h (x) d x ≤ 1 K(ε) < ε −π µ2 if µ is large enough, K(ε) being a number independent of µ. The lemma follows by the genericity of ε and (D.10). □ For µ → ∞, the first term on the right hand side of (D.7) can be rewritten expanding the square root as 1 s h µ − V1 (x) p(x) 1 µ2 V1 (x) −2 = p 1− +O µ 2µ h p(x) 1 µ2 V1 (x) 1 . = p − +O 1 p 3 h p(x) 2hµ 2 p(x) hµ 2 Then, in the case p(x) = 1, θ′h (x, µ) = 1Æ 1 V1′ (x) sin(2θ h (x, µ)), µ − V1 (x) − h 4 µ − V1 (x) (D.11) and asymptotically as µ → ∞ the first term on the right hand side becomes 1Æ h 1 µ − V1 (x) = µ2 h − V1 (x) 1 2hµ 2 +O 1 3 . (D.12) hµ 2 Let µn for n ∈ N denote the eigenvalues of the Sturm-Liouville periodic problem (D.1). We know from Appendix A that they exists, are discrete, at most doubly degenerate and accumulate at infinity. Then we have the following theorem. Theorem D.3. Let p(x) = 1. Then as m → ∞, µ2m+1 and µ2m+2 both satisfy p ∫π µ = (m + 1)h + −π V (x)d x 4π(m + 1)h +o 1 mh . Proof. Fix an ε > 0. Le V1 (x) be a continuously differentiable function with period 2π such that V1 (x) ≥ V (x) (D.13) 118 D. GENERALIZED PRÜFER TRANSFORMATION IN THE SEMI-CLASSICAL LIMIT and ∫ π −π ∫ V1 (x)d x ≤ ε + π −π V (x)d x. (D.14) Let µ1,n denote the eigenvalue in the periodic problem associated with V1 (x) (and with p(x) = 1) and ψ1,n its eigenfunction. Then by Theorem A.8 and (D.13) we have µ1,n ≥ µn . We can assume that ψ1,n (−π) ≥ 0 and we apply the modified Prüfer transformation to y(x) = ψ1,2m+1 (x) with a0 = −π in (D.3). Now, from (D.3) and (D.9) we have 2kπ ≤ θ(π, µ1,2m+1 ) < (2k + 1)π for some integer k . For Theorem A.6 we know that ψ1,2m+1 has 2(m + 1) zeroes in (−π, π], hence by (D.5) with a1 = π we have 2k = 2(m + 1) and thus θ h (π, µ1,2m+1 ) − θ(−π, µ1,2m+1 ) = 2(m + 1)π. (D.15) Integrating (D.11) with µ = µ1,2m+1 over [−π, π] we obtain ∫ 2(m+1)π = π −π 1Æ h µ − V1 (x)d x− 1 4 ∫ π V1′ (x) −π µ − V1 (x) sin(2θ h (x, µ))d x. (D.16) By Lemma D.2 the rightmost term is o(µ−1 ) as m → ∞ (becoming o(h/m 2 ) in (D.17) and thus being suppressed from the equation). For the first integral on the right we can use the binomial expansion as in (D.12). Thus (D.16) gives ∫π 1 2(m + 1)π = µ2 h 2π − that is 1 p µ − (m + 1)h µ − 4π −π V1 (x)d x 1 2hµ 2 ∫ π −π +O V1 (x)d x + O 1 3 hµ 2 1 µ = 0. Solving for µ one gets s ∫π 1 1 p µ = (m + 1)h + (m + 1)2 h 2 + V1 (x)d x + O(µ−1 ) . 2 π −π Extracting (m + 1)h and using once more the binomial expansion one gets p ∫π µ2m+1 = (m + 1)h + −π V1 (x)d x 4π(m + 1)h +O 1 m2 h 2 . (D.17) Hence by (D.13), (D.14) and by the fact that ε is arbitrarily small p ∫π µ2m+1 ≤ (m + 1)h + −π V (x)d x 4π(m + 1)h +o 1 mh . The opposite inequality can be proved in the same way. The result for µ2m+1 holds in the same form using the fact that its eigenfunction must have 2(m +1) zeroes. □ D. GENERALIZED PRÜFER TRANSFORMATION IN THE SEMI-CLASSICAL LIMIT 119 So far we have not used any differentiability-related property of V . Using the differentiability, we can improve incredibly the previous result. Theorem D.4. Let p(x) = 1, let r ∈ N, and let piecewise continuous. Then µ2m+1 and µ2m+2 both satisfy p µ = (m +1)h + r +1 ∑ Ak k=1 (m + 1)k h k +O dr V (x) dxr 1 exist and be +o m r +2 h r +2 1 m r +1 h r −2 where the Ak are independent of m and involve q(x) and its derivatives up to order r − 1. In particular, A1 = ∫ 1 4π π −π V (x)d x, A2 = 0, A3 = ∫ 1 16π π −π V 2 (x)d x − A21 . (D.18) Proof. We consider V1 (x) = V (x) in (D.11), then µ1,n = µn and the case r = 1 correspond simply to (D.17). To deal with r ≥ 2 we reconsider (D.16), which is now ∫ 2(m + 1)π = 1Æ π h −π µ − V (x)d x − 1 ∫ 4 π V ′ (x) −π µ − V (x) sin(2θ h (x, µ))d x (D.19) and µ is µ2m+1 or µ2m+2 . By (D.11), with V1 (x) = V (x), the second integral on the right in (D.19) is ∫ π π hV ′ (x) (µ − V (x)) θ′h (x, µ) + 3 2 = h ∫ 2 + − π −π h ∫ 8 h ! V1′ (x) 4 µ − V1 (x) V ′ (x) d d x (µ − V (x)) 32 π V ′2 (x) −π (µ − V (x)) 2 π V ′2 (x) −π (µ − V (x)) 2 ∫ 8 1 sin(2θ h (x, µ)) sin(2θ h (x, µ))d x cos(2θ h (x, µ))d x 5 dx 5 cos(4θ h (x, µ))d x (D.20) 3 after integrating by parts. The first term on the right here is o hµ− 2 D.2, the last is o hµ − 52 ∫π 1 2(m + 1)π = h 2π − by Lemma 5 for the same reason and the central one is O hµ− 2 . This, together with the binomial expansion of right of (D.19) gives µ2 −π V (x)d x 1 2hµ 2 ∫π − −π p µ − V (x) in the first term on the V 2 (x)d x 3 8hµ 2 +O 1 5 hµ 2 +o h 3 µ2 (D.21) 120 D. GENERALIZED PRÜFER TRANSFORMATION IN THE SEMI-CLASSICAL LIMIT 1 To solve (D.21) for µ 2 in terms of m , we write it as 1 2 µ = M +µ − 12 A1 + µ − 32 (A3 − A21 ) + O 1 m5 h 5 +o 1 m3 h (D.22) where M = h(m + 1). Then, taking the reciprocals we obtain 1 1 µ− 2 = M −1 1 − µ− 2 A1 M −1 + O(h −4 m −4 ) = M −1 − M −3 A1 + O(h −5 m −5 ). (D.23) And thus, 3 µ− 2 = M −3 + O(h −5 m −5 ). 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