Analytic properties of the Jost functions

Transcription

1 Anaytic properties of the Jost functions by Yannick Mvondo-She Submitted in partia fufiment of the requirements for the degree Magister Scientiae in the Facuty of Natura and Agricutura Sciences University of Pretoria Pretoria Apri 213

2 i Decaration I, the undersigned, hereby decare that the dissertation submitted herewith for the degree Magister Scientiae to the University of Pretoria contains my own, independent work and has not been submitted for any degree at this or any other university. Signature: Name: Date:

3 ii Acknowedgments Remember how the Lord your God ed you a the way in the desert these forty years, to humbe you and to test you in order to know what was in your heart, whether or not you woud keep his commands. He humbed you, causing you to hunger and then feeding you with manna, which neither you nor your fathers had known, to teach you that man does not ive on bread aone but on every word that comes from the mouth of the Lord. Your cothes did not wear out and your feet did not swe during these forty years. Know then in your heart that as a man discipines his son, so the Lord your God discipines you. Observe the commands of the Lord your God, waking in his way and revering him. For the Lord your God is bringing you into a good and-a and of streams and poos of water, with springs fowing in the vaeys and his. (Deuteronomy 8:2-7) Heaveny Father, thank you for, through the good and the bad, you aways stood by me, and aowed me to bring this work to competion. Bessed be your name. I am deepy indebted to my supervisor Professor P. Seyshchev and to my co-supervisor Professor S.A. Rakitiansky, for having invoved me in such an interesting project. Thank you for the financia support, for your patience and for the freedom you gave me whie working on the project. My sincere appreciation is expressed to the Pretoria East Church Connect Group for bessing me with the Word of God every Thursday evening and for spiritua growth. To my famiy in Cameroon, in France and in the french West Indies, thank you very much for your ove and your encouragement. My ast word of thanks goes to my mother, to whom I owe a great debt of gratitude for a ife of sacrifice: Maman, à travers ce petit ouvrage que je te dédie, peut être pouvons nous nous inspirer de Romains 8:18, et nous dire que es souffrances du temps présent (et surtout passé) ne sauraient être comparabe à a goire qui nous sera révéée. Merci beaucoup Maman pour tout ce que tu as fait pour moi.

4 iii Tite: On some anaytic properties of the Jost functions Student: Yannick Mvondo-She Supervisor: Professor Pave Seyshchev Co-supervisor: Professor Sergei Rakitianski Department: Physics Degree: MSc Abstract Recenty, was deveoped a new theory of the Jost function, within which, it was spit in two terms invoving on one side, singevaued anaytic functions of the energy, and on the other, factors responsibe for the existence of the branching-points. For the singe-vaued part of the Jost function, a procedure for the powerseries expansion around an arbitrary point on the energy pane was suggested. However, this theory acks a rigorous proof that these parts are entire functions of the energy. It aso gives an intuitive (not rigorous) derivation of the domain where they are entire. In the present study, we fi this gap by using a method derived from the method of successive approximations. Résumé Récemment, une nouvee théorie sur es fonctions de Jost a été déveoppée, dans aquee, es fonctions de Jost sont divisée en deux parties, avec d une part des fonctions uniformes (univauées et anaytiques) de energie, et d autre part, des facteurs responsabes de existence de points de ramification. Une procédure permettant e déveoppement en série de a partie contenant es fonctions anaytiques de énergie autour d un point queconque du pan compexe de énergie a notamment été suggérée. Cependant, cette théorie souffre d une preuve rigoureuse de anayticité de ces fonctions. La théorie permet égaement d obtenir, à encore de facon intuitive, e domaine d anayticité de ces fonctions. Nous nous proposons donc, à aide d une méthode dérivée de cee des approximations successives de démontrer que ces fonctions sont anaytiques dans un domaine particuier que nous déterminerons de facon expicite.

5 List of Figures 2.1 Function f(x) Path taken by z The energetic Riemann surface Domain of expansion of the anaytic function f(z) in power series Anaytic continuation of f(z) aong a path Rectanguar domain R surrounding the point (x, y ) Domain of anayticity of the functions Ã(E, ) and B (E, ) 53 iv

6 Contents 1 Introduction Historica background The Jost function Basic concepts Boundary conditions Transformation of the Schrödinger equation Factorization Anaytic properties of the Jost function Mathematica background Functions of a compex variabe Anaytic function Singe-vaued and many-vaued functions Riemann surfaces Factorization revisited Properties of anaytic functions Existence and nature of soutions of ordinary differentia equations Background The existence theorem The method of successive approximations Gronwa inequaity Gronwa Lemma [34] On certain methods of successive approximations A genera theorem on inear differentia equations that depend on a parameter Extension of the method of successive approximation to a system of differentia equations of the first-order; vector-matrix notation A gance at existence and uniqueness Appication to inear equations Vector-matrix notation Norms of matrices v

7 CONTENTS An inequaity invoving norms and integras Anayticity of the functions à (E, r) and B (E, r) Existence and uniqueness Anayticity Anayticity of the functions à (E, ) and B (E, ) Asymptotics of the Ricatti-Besse and of the Ricatti-Neumann functions The integra equation and its Kerne Restriction on the Kerne Concusion 54

8 Chapter 1 Introduction 1.1 Historica background By the past, as a conventiona way to treat quantum coision processes, common practice was to focus on the scattering ampitude of the physica wave function [1] [2]. Yet, the anaysis of non reativistic quantum mechanica probems can be done adequatey in terms of the Jost functions, and the Jost soutions of the Schrödinger equation. The Jost function was introduced in 1947 by Res Jost [3]. It can be described in substance as a compex function of the tota energy of a quantum state, where the energy is aowed to have not ony rea but aso compex vaues [4]. The Jost functions, when defined for a compex vaues of the momentum possess a information about a given physica system. An interesting feature in the Jost function approach is that it aows a simutaneous treatment of bound, virtua, scattering and resonance states. Abundant itterature on Scattering Theory has chapters devoted to the Jost function, where usuay it is expressed either via an integra containing the reguar soution [1] [2], or via a Wronskian of the Jost soutions [5]. In a cases, they are expressed in terms of the wave function. But to make use of the Jost functions in such a form, one must find the wave function first. This means that the probem is practicay soved and nothing more is needed [6]. Hence in spite of the usefuness of the Jost functions in studying spectra properties of hamitonians, for quite some time they were regarded rather as purey mathematica entities, eegant and usefu in forma scattering theory [7], but with no computationa use. In the eary nineteen nineties, inear first-order differentia equations for functions cosey reated to the Jost soutions were proposed [8]. Based on the variabe-phase approach [9], the equations and their soutions provide the Jost function at any fixed vaue of the radia variabe r, and its compex 2

9 CHAPTER 1. INTRODUCTION 3 conjugate counterpart, which corresponds to the potentia truncated at the point r. An inconvenience though, was that the method was suitabe ony for bound and scattering state cacuations, meaning, for cacuations in the upper-haf of the compex pane and on the rea axis. An extension of the method to the unphysica sheet was proposed in [1], in order to incude the resonant state region. Such deveopment was made by combining the variabe constant method [11], and the compex coordinate rotation method [12]. As a resut, the combination used to recast the Schrödinger equation into a set of inear first ordered couped equations for Jost type soutions aowed for a treatment of a possibe states in a unified way. Concusive tests confirmed the effectiveness of the approach, in particuar, in ocating bound states and resonances, through numerica integrations of the derived equations in order to obtain the Jost functions (Jost matrices in the case of mutichannes) for a momenta of physica interest. Athough the method enjoyed success, it aso had drawbacks. One of them concerned the point k = at which the proposed equations are singuar. The method was thereafter refined by using a procedure taken from [13]. The prescription ies in the fact that within a sma region around k =, the Ricatti-Hanke functions h (±) (kr) can be expanded in power series [14], and each term therein can be factorized in k and r. Simiary, the Jost functions can be expanded in powers of k with unknown r dependent coefficients in this region, the coefficients being specified by the resuting system of k independent differentia equations. This is of a crucia importance in fieds such a Quantum few-body Theory, especiay when it comes to ocating quantum resonances... Historicay, after the advent of Quantum Mechanics, attention to resonance states was first drawn by nucear physicists. In particuar, we can think of George Gamow s semina work on the α-decay. The roe of quantum resonances in Soid State and Chemica Physics for instance was understood much ater [15] [16]. More recenty, efforts in a Condensed Matter and Moecuar Physics oriented research have converged to construct the soutions of the set of first order differentia equations in the form of Tayor-type power series near an arbitrary point on the Riemann surface of the energy [17], in a way that is simiar to the effective expansion range, but more generaized. A fundamenta point in the expansion of a function, is the anaycity of the given function. For a the aforementionned work, it was given that the Jost function is anaytic at a compex energies and that for the so caed spectra points, it has simpe zeros. The spectra points being the energies

10 CHAPTER 1. INTRODUCTION 4 at which the system forms bound and resonant states, the ocation of the bound states and the resonances is done by cacuating the Jost function and the points of its zeros. Yet, the anaytic properties of the Jost functions suffers a ack of rigorous treatment. A specia attention to the probem wi be given here. 1.2 The Jost function Basic concepts A range of macroscopic phenomena can be described on the basis of nonreativistic Quantum Mechanics. Moecuar, Atomic, Nucear and Soid State phenomena span this range. At a given time t, the state of a physica system can be described within the framework of non-reativistic Quantum Mechanics, by a compex-vaued wave function Ψ( r, t), where the wave function Ψ depends on the timeparameter t, and on a compete set of variabes summarized as r. The physica system can usuay be found in a quantum state, which is characterized by a fu set of quantum numbers, such as tota energy, anguar momentum, etc,... The hermitian operator Ĥ that describes the energy of a system is the Hamitonian. It consists of the kinetic energy operator T = N i=1 p 2 i 2m i, (1.1) of a system of N spiness partices of mass m i and momentum p i, of a potentia energy operator V, which is in genera the function of the N interpartices dispacement vectors (pus a potentia generated by an externa fied, if any) Ĥ = T + V. (1.2) The hamitonian of a physica system determines its evoution in time. In coordinate representation, the evoution of a state is described by a partia differentia equation, the Schrödinger equation ĤΨ( r, t) = iħ Ψ( r, t). (1.3) t For a time-independent Hamitonian Ĥ, the wave function Ψ( r, t) = exp ( iħ ) Et ψ ( r) (1.4)

11 CHAPTER 1. INTRODUCTION 5 is said to be a soution of the Schrödinger equation (1.3), if and ony if ψ ( r) is an eigenfunction of Ĥ, with eigenvaue E, such that Ĥψ( r) = Eψ ( r). (1.5) Equation (1.5) is caed the time-independent, or stationary Schrödinger equation. For a point partice in a radiay symmetric potentia V ( r), in coordinate representation, the time-independent Schrödinger equation is ) ( ħ2 2µ r + V (r) ψ( r) = Eψ( r). (1.6) It is possibe with the hep of the orbita anguar momentum L, to express the Lapacian operator = = p 2 in spherica coordinate x 2 y 2 z 2 ħ 2 = 2 r r 2 r + L r 2 ħ 2. (1.7) The square and the z component of the anguar momentum, respectivey L 2 and L z being constants of motion, the soutions of the Schödinger equation (1.5) can be abeed by the good quantum numbers and m, and the energy E, to give [1] [2] [5] ψ( r) = φ (E, r)y,m (θ, ϕ). (1.8) In equation (1.8), the so-caed spherica harmonic function Y,m (θ, ϕ) is an eigenfunction of both operators L 2 and Lz, whie and m are eigenvaues of L 2 and L z respectivey. The soutions of the stationary Schrödinger hence have a radia part in φ (E, r) and an anguar part from the spherica harmonic function expressed in terms of θ and ϕ, the poar anges of r. Inserting (1.8) into (1.6) eads to an equation for the radia wave function φ (E, r) [ ħ ( d 2 2µ dr ) ] d ( + 1)ħ2 + r dr 2µr 2 + V (r) φ (E, r) = Eφ (E, r), (1.9) that is independent of the azimuta quantum number m. The ordinary differentia equation of second order for the radia wave function φ (E, r) (1.9) is caed the radia Schrödinger equation. It provides a

12 CHAPTER 1. INTRODUCTION 6 significant simpification of the partia differentia equation (1.6). Yet, with a itte bit more agebra a greater simpification can be obtained by formuating an equation not for φ (E, r), but for u = rφ, i.e for the radia wave function of u (r) defined by ψ( r) = u (E, r) Y,m (θ, ϕ). (1.1) r We end up with the foowing expression of the radia Schrödinger equation ( ħ 2µ d 2 ) ( + 1)ħ2 + dr2 2µr 2 + V (r) u (E, r) = Eu (E, r). (1.11) Equation (1.11) is actuay the Schrödinger equation for a singe partice of mass µ moving in a one spatia dimension, in an effective potentia consisting of V (r) pus the centrifuga potentia ( + 1)ħ 2 \2µr 2 V eff (, r) = V (r) Boundary conditions ( + 1)ħ2 2µr 2. (1.12) The radia Schrödinger equations (1.9) and (1.11) are defined ony for nonnegative vaues of the radia coordinate r, i.e on the interva r [, ). The boundary condition imposed on the radia wave function u (E, r) at r =, can be derived by inserting an ansatz u (E, r) r α into equation (1.11). As ong as the potentia V (r) is ess singuar than r 2, the eading term on the eft-hand side of the ansatz is proportiona to r α 2, and vanishes ony if α = +1 or α = [18]. The atter option must be discarded, for an infinite vaue of u (E, r ) woud ead to an infinite contribution to the norm of the wave function near the origin; a finite vaue woud ead to a deta function singuarity coming from ( ) 1 r in equation (1.6), which cannot be compensated by any other term in the equation. The boundary condition imposed at r = on the radia wave function is thus u (E, ) =, for a. (1.13) The behaviour of the radia wave function near the origin is given by u (E, r) r +1, for r (1.14) (for potentias that are ess singuar than r 2 ). These conditions remain the same for a types of soutions. However, when

13 CHAPTER 1. INTRODUCTION 7 the radia coordinate tends to an infinite vaue, the boundary condition is different for the bound, scattering and resonant states. In particuar, when the potentia V (r) at arge distances is ess singuar than r 2, and therefore vanishes fast enough, the radia Schrödinger equation (1.11) becomes ( d 2 dr 2 + k2 ) ( + 1) r 2 u (E, r) =, for r, (1.15) where k is caed the wave number and is reated to the energy by 2mE = 2 k 2 (and therefore k = p, p being the momentum). Equation (1.15) is aso caed the free radia Schrödinger equation. Its genera soution can be constructed as the inear combination of two ineary independent soutions h (±) (kr). These soutions are the Ricatti-Hanke functions, which behave exponentiay h (±) (kr) r ie [±i(kr π 2 )], (1.16) and the genera asymptotic form of u (E, r) reads u (E, r) r ah ( ) (kr) + bh (+) (kr), (1.17) where a and b are arbitrary compex numbers that by an appropriate combination determine the soution type (bound, scattering or resonant). In fact, these coefficients are compex functions of the tota energy of the system and differ with the anguar momenta. Equation (1.17) can then be rewritten as u (E, r) r h ( ) (kr)f (in) (E) + h (+) (kr)f (out) (E), (1.18) with a = f (in) (E) and b = f (out) (E). These two functions are caed the Jost functions. Since h ( ) (kr) represents the incoming spherica wave, and h (+) (kr) the outgoing spherica wave, these two functions are just the ampitudes of the corresponding waves. 1.3 Transformation of the Schrödinger equation At stake here is to show the anaytic properties of the Jost function. This is equivaent to expressing it in such a way that a possibe terms that are not anayticay dependent on the energy are given expicity. This can be done by transforming the radia Schrödinger equation (1.9) into simpe differentia equations of the first order. The radia Schrödinger equation reads

14 CHAPTER 1. INTRODUCTION 8 ( d 2 dr 2 + k2 ) ( + 1) r 2 u (E, r) = V (r)u (E, r). (1.19) Using a systematic method taken from the theory of Ordinary Differentia Equations, and known as the method of Variation of Parameters [11] [19], we ook for the unknown function u (E, r) in specia form u (E, r) = h ( ) (kr)f (in) (E, r) + h (+) (kr)f (out) (E, r), (1.2) where F (in) (E, r) and F (out) (E, r) are new unknown functions. This impies that, with the Jost functions as a case of interest, we wi ook for an asymptotic-ike soution, for it is known that at arge distances, where the potentia vanishes, the wave function behaves as a inear combination of the Ricatti-Hanke functions obeying the equation ( d 2 dr 2 + k2 ) ( + 1) r 2 u (E, r) =. (1.21) So at arge distances, these functions are constant. As indicated in [1], the introduction of two unknown functions F (in/out) instead of the origina unknown function u impies they cannot be independent. Therefore, an arbitrary condition reating them to each other can be imposed. Convenienty, using r as d dr, the foowing equation can be chosen h ( ) (kr) r F (in) (E, r) + h (+) (kr) r F (out) (E, r) =. (1.22) This condition is known as the Lagrange condition in the method of Variation of Parameters. Substituting equation (1.2) into equation (1.19), and using the Lagrange condition and the Wronskian of the Ricatti-hanke function h ( ) (kr) r h (+) (kr) + h (+) (kr) r h ( ) (kr) = 2ik, (1.23) yieds a couped system of first order differentia equations for the unknown functions, that are nothing ese but an equivaent form of the origina Schrödinger equation [1] r F (in) (E, r) = h(+) (kr) 2ik r F (out) (E, r) = h( ) (kr) 2ik [ V (r) [ V (r) h ( ) h ( ) (kr)f (in) (E, r) + h (+) (kr)f (in) (E, r) + h (+) ] (kr)f (out) (E, r) ] (kr)f (out) (E, r)

15 CHAPTER 1. INTRODUCTION 9 Whie trying to find the boundary conditions that shoud be imposed on the functions F (in/out) (E, r), it must be remembered with the restrictions on the potentia, that the physica soution u (E, r) must be reguar everywhere. This impies [17] that the wave function u (E, r) must be zero when r is zero u (E, r) r, (1.24) and is proportiona to the Ricatti-Besse function at short distances u (E, r) j (E, r). (1.25) It coud be argued that it is not the case because h (+) (kr) and h ( ) (kr) in equation (1.2) are singuar at r =. But their singuarities can cance each other if they are superimposed with a same coefficient [2] The condition h (+) + h ( ) = 2j (kr). (1.26) u (E, ) =, (1.27) can ony be achieved if both F (in/out) (E, r) are equa to the same constant at r = F (in) (E, ) = F (out) (E, ). (1.28) Since we are not concerned about their normaization, we chose any arbitrary vaue for the constant. We chose the constant to be 1 2 for u (E, r) to behave near the origin as the Ricatti-Besse function, as prescribed by equation (1.26) F (in) (E, ) = F (out) (E, ) = 1 2. (1.29) In order to express the non-anaytic dependencies of the Jost functions in an expicit form, the ansatz (1.2) can be recast by using either the Ricatti- Besse and Ricatti-Newmann functions, j (kr) and y (kr), or the Ricatti- Hanke functions, that are reated by h (±) (kr) = j (kr) ± iy (kr). (1.3)

16 CHAPTER 1. INTRODUCTION 1 This eads to another representation of the physica soution of equation (1.19) in the form u (E, r) = j (E, r)a (E, r) y (E, r)b (E, r), (1.31) equivaent to the ansatz (1.2), not ony at arge distances, but everywhere on the interva r [, ). When r, the functions A (E, r) and B (E, r), and simiary F (in/out) (E, r) tend to r-independent constants. In particuar, the asymptotic behaviour of the wave function eads to u (E, r) r h ( ) (kr)f (in) (E) + h (+) (kr)f (out) (E), (1.32) where, when comparing equation (1.32) and equation (1.2), we see that F (in/out) (E, r) converge to the Jost functions when r and im F (in) r (E, r) = f (in) (E), (1.33) im F (out) r (E, r) = f (out) (E). (1.34) The unknown functions A (E, r) and B (E, r) can be expressed in terms of F (in/out) (E, r), by using equation (1.3), and making a inear combination of the system between equation (1.23) and equation (1.24), eading to and asymptoticay to A (E, r) = F (in) (E, r) + F (out) (E, r) [ B (E, r) = i F (in) A (E) = f (in) [ B (E) = i f (in) ] (E, r) F (out) (E, r). (E) + f (out) (E) ] (E) f (out) (E). (1.35) (1.36) From (1.36), a new expression of F (in/out) (E, r) is found as F (in) (E, r) = 1 2 [A (E, r) + ib (E, r)] F (out) (E, r) = 1 2 [A (E, r) ib (E, r)], (1.37)

17 CHAPTER 1. INTRODUCTION 11 and from which the Jost functions can be obtained, considering the asymptotic behaviour of (1.37), by writing f (in) (E) = 1 2 [A (E) + ib (E)] f (out) (E) = 1 2 [A (E) ib (E)]. (1.38) A new system of firt order differentia equations equivaent to the system between equations (1.23) and (1.24), and to equation (1.19) is obtained for the new unknown functions A (E, r) and B (E, r), and reads [2] r A (E, r) = y (kr) V (r) [j (kr)a (E, r) y (kr)b (E, r)] k r B (E, r) = j ] (kr) V (r) [j (kr)a (E, r) y (kr)b ( k E, r). (1.39) Foowing from equation (1.29), the system has the physica boundary conditions A (E, ) = 1, B (E, ) =. (1.4) What was showed here is a simpe procedure to express the Jost functions for finite range potentias. For such potentias, when arge enough vaues of r are reached, or if the potentia is cut off at a certain (arge) vaue of r, the functions do not change anymore, and eventuay give us the Jost functions Factorization The main goa being to show that the Jost function can be spit into two parts, one of which is anaytic and can therefore be expressed in power-series expansion, a further transformation of equation (1.39) is required to expicity separate the non-anaytic factors. Using the fact that the Ricatti-Besse and Ricatti-Neumann functions can be represented by absoutey convergent series [2] j (kr) = y (kr) = ( ) kr +1 2 n= ( ) 2 kr n= ( 1) n π Γ ( n) n! ( 1) n++1 Γ ( n) n! ( ) kr 2n = k +1 j (E, r) 2 (1.41) ( ) kr 2n = k ỹ (E, r). 2

18 CHAPTER 1. INTRODUCTION 12 The factorized functions j (E, r) and ỹ (E, r) are obtained. These functions have the advantage that they do not depend on odd powers of k, and thus are singe-vaued functions of the energy E. Using equation (1.41) [2], it is possibe to express A (E, r) and B (E, r) as a inear combination of products momenta factors and new functions Ã(E, r) and B (E, r), such that equation (1.39) can be transformed. If we write j (kr) = k +1 j (E, r) y (kr) = k ỹ (E, r), equation (1.39) becomes r A (E, r) = k ỹ (kr) [ V (r) k +1 j ] (kr)a (E, r) k ỹ (kr)b (E, r) k r B (E, r) = k+1 j (kr) V (r) [k +1 j ] (kr)a (E, r) k ỹ (kr)b ( k E, r), or again ] r A (E, r) = [ j ỹ(kr) k +1 V (r) (kr)k +1 A (E, r) ỹ (kr)k B (E, r) r B (E, r) = k j (kr)v (r) [ j (kr)k +1 A (E, r) ỹ (kr)k B (E, r) ], which in turn gives k +1 r A (E, r) = ỹ (kr)v (r) [ j (kr)k +1 A (E, r) ỹ (kr)k B (E, r) ] k r B (E, r) = j (kr)v (r) [ j (kr)k +1 A (E, r) ỹ (kr)k B (E, r) ], or r [ k +1 A (E, r) ] = ỹ (kr)v (r) [ j (kr)k +1 A (E, r) ỹ (kr)k B (E, r) ] r [ k B (E, r) ] = j (kr)v (r) [ j (kr)k +1 A (E, r) ỹ (kr)k B (E, r) ], and if we write à (E, r) = k +1 A (E, r) and B (E, r) = k B (E, r), (1.42) we finay have an expression

19 CHAPTER 1. INTRODUCTION 13 r à (E, r) = ỹ (E, r)v (r) [ j (E, r)ã(e, r) ỹ (E, r) B ] (E, r) r B (E, r) = j (E, r)v (r) [ j (E, r)ã(e, r) ỹ (E, r) B ] (1.43) ( E, r), devoid of a momenta factors, where remains a system of first order differentia equations, whose coefficients and soutions are singe-vaued functions of the energy E Anaytic properties of the Jost function Up to this point we have expressed the system of first-order differentia equations between equations (1.23) and (1.24) by a new set of first-order differentia equation (1.43). The anaytic properties of the Jost functions that we are trying to estabish, are subject to the proof that for any r on the interva [, ), the soutions of equation (1.43), namey Ã(E, r) and B (E, r), are entire (anaytic singe-vaued) functions of the compex variabe E. The approach for this is to use a theorem taken from a treatise pubished in 1896 by french mathematician Emie Picard ( ), based on a theory caed the Method of Approximations. This method, athough probaby known to Cauchy, originates in 1838 when Joseph Liouvie appied it to the case of the homogeneous inear equation of the second order [21]. The theory was extended to inear equations of order n by J. Caqué in 1864 [22], L. Fuchs in 187 [23] and G. Peano in 1888 [24], but in its most genera form (incuding non inear differentia equations), it was deveoped by Picard in 1893 [25]. In particuar, is found in the treatise a theorem that states [26] the foowing: Let a inear differentia equation of the form: d n y dx n + P 1(x, k) dn 1 y dx n P n(x, k)y = Q(x, k), (1.44) where P i (x, k) (with i = 1, 2,..., n) and Q(x, k) are continuous functions of the rea variabe x and singe-vaued anaytic functions of the compex parameter k. The method of successive approximations shows that there exists a unique soution (rea or not), with given initia conditions (rea or not), and that if these initia conditions are independent of k, the soution is aso an anaytic function of k. This resut was aso proved by Poincaré using a different method [27]. The theorem aso mentions that for a system where Q(x, k) =, the system of soutions formed must be independent of k -and in fact must assume numerica vaues- at the initia conditions. The appication of this theorem is the object of the next chapters.

20 Chapter 2 Mathematica background We wi discuss here, some topics that are of specific interest to us. In this view, this chapter shoud merey be regarded as a too-box. 2.1 Functions of a compex variabe In trying to investigate differentia equations with the hep of power series, one reaises quicky that a knowedge of the theory of functions of a compex variabe is needed. Here, we wi confine our deveopment to the part of Compex Anaysis that wi be usefu to our present study of the anaytic properties of the soutions of the system of inear differentia equations under consideration. The basic properties of Compex Functions can be found in [11] Anaytic function Consider a compex variabe z = x + iy, where x and y are independent rea variabes. z usuay represents a point in a compex z-pane. Let f(z) = u + iv, a function of the compex variabe z be defined by associating to each point z a given compex number f(z). The function f(z) is caed a singe-vaued anaytic function of z, if u and v are rea singe-vaued functions of x and y. In Rea Anaysis, f(x) is usuay defined as a function of a rea variabe x. When f(x) has a derivative, then the quotient f(x + h) f(x) h approaches f (x) when h approaches zero. (2.1) 14

21 CHAPTER 2. MATHEMATICAL BACKGROUND 15 y f(x+h) f(x) x x+h Figure 2.1: Function f(x) x In the same way, in Compex Anaysis, if we write z = z z and f(z) = f(z) f(z ) = f(z + z) f(z ), it is possibe to determine under which conditions the quotient f(z) z wi approach a definite imit when the absoute vaue of z approaches zero. By etting x and y have independent increments x and y, z wi be incremented by z = x + i y. If f(z) is a singe-vaued function of z, f(z) wi receive an increment f(z) = u + i v. The derivative of f(z) with respect to z can then be expressed as df(z) dz f(z) u + i v = im = im z z ( x, y) (,) x + i y, (2.2) where the imit, if it exists, must have a singe vaue, which is independent of the path taken by z (or x and y) to approach zero. This means that the imit in equation (2.2) is a doube imit with respect to the increments x and y. Im z z + z y z x Figure 2.2: Path taken by z Re z The existence of a doube imit impies that the corresponding iterated imits aso exist and are equa. Let x and y in figure (2.2) approach zero in the foowing way: first y, then x. Letting z in that way aows to write

22 CHAPTER 2. MATHEMATICAL BACKGROUND 16 dw dz = im x y = im x = im x im u + i v x + i y u + i v x u x + i im x v x. Therefore, if the derivative exists, it wi have the vaue dw dz = u x + i v x. (2.3) Simiary, if the imit in equation (2.2) exists, it can identicay be evauated by etting x first and then y. Thus dw dz = im y x = im y = i im y im u + i v x + i y u + i v i y u y + im y and if the derivative exists, it has the vaue v y, dw dz = i u y + v y. (2.4) Now, if the derivative exists throughout some region incuding the point z, equations (2.3) and (2.4) must then be identica in that region, u and v being rea functions of the rea variabes x and y, it is possibe to equate rea and imaginary parts in the equations (2.3) and (2.4). This yieds u x + i v x = i u y + v y. (2.5) Then, if the first derivatives of u and v with respect to x and y are continuous at a point, a necessary and sufficient condition for the existence of the derivative at the given point is that u and v satisfy the Cauchy-Riemann equations u x = v (2.6) y u = v y x, (2.7)

23 CHAPTER 2. MATHEMATICAL BACKGROUND 17 throughout some neighbourhood of the point. From this, it can be inferred that a function is anaytic at the point z = z if and ony if the above derivative exits at each point in some neighbourhood of the point. A function that is anaytic at every point of a region is said to be anaytic in that region Singe-vaued and many-vaued functions In defining the anaytic function of a compex variabe z, we have considered a function f(z) that has assigned to it a definite vaue for each point z of a connected region, such that f(z) has a continuous derivative in the region. These conditions ed to the definition of a singe-vaued anaytic function of z in the domain. It foows that e z, cos z, sin z, cosh z, sinh z, (2.8) are singe-vaued anaytic functions in the entire pane, as they a have continuous derivative for any z [28]. In the same way, 1 z 2 1 (2.9) for instance, is a singe-vaued anaytic function in a region formed by the whoe z-pane except the zeros of the denominator z = ±1, and tan z is singe-vaued anaytic on a region formed by the whoe pane except the infinite point set... ; 7 π 2 ; 5π 2 ; 3π 2 ; π 2 ; π 2 ; 3π 2 ; 5π 2 ;... (2.1) Suppose now that f(z) has in genera more than one vaue assigned to it for the points of the region. f(z) is then said to be a many-vaued anaytic function if its vaues can be grouped in branches, each of which is a singevaued anaytic function about each point of the region [28]. To understand the nature of a many-vaued function, we can use the fact that an anaytic function is necessariy continuous at a points at which it is anaytic. Using a geometricay suggestive approach, the idea of continuity can be expressed by taking into consideration the fact that the vaue of a function of the variabe z does not aways depend entirey upon the vaue of z aone, but to a certain extent aso upon the successive vaues assumed by the z, when going from the initia vaue to the actua vaue in consideration [32], in other words, upon the path taken by the variabe z. This eads to a new definition: an anaytic function f(z) is said to be singevaued in a region when a paths in that region which go from a point z

24 CHAPTER 2. MATHEMATICAL BACKGROUND 18 to any other point z ead to the same fina vaue for f(z). When on the other hand, the fina vaue of f(z) is not the same for a possibe paths in the region, the function is said to be many-vaued. An important statement in connection with this is found in [32]: a function that is anaytic at every point of a region is necessariy singe-vaued in that region. Now, if the path from z to z describes a cosed circuit, and we return to our point of departure after having gone through the path, there are two possibiities: either we arrive again at the same vaue of the function, and we there have a necessary and sufficient condition for a function to be singevaued anaytic at every point of a region, or we do not. In this case, we have many-vaued function and f(z ) wi have at east two different meanings. As an exampe, we wi consider the ogarithmic function og z. Using poar coordinates r, θ, it can be shown [3] that og z = og r + i(θ + 2πn), n =, ±1, ±2,... (2.11) It is easy to verify that this expression satisfies the Cauchy-Riemann equations, and therefore that og z as expressed in equation (2.11) is an anaytic function of z. The foowing equation d dz {og z} = 1 z (2.12) shows that the derivative of og z is not defined for z =. The origin is thus a point where the derivative of the function and the function itsef cease to be continuous. It is caed a singuar point of the function. Equation (2.11) seems to impy that there exists an infinite number of different ogarithmic functions, each of them having a different vaue of n. In reaity, they are a branches of one and the same function, and the integer n merey accounts for the function to be many-vaued. Indeed, et the vaue of n be arbitrariy taken as zero, and et z move in the positive direction aong the circe z = r, starting from the point (r, ). og r wi remain constant and θ wi grow continuousy. When z wi return to its origina position, the function og z wi therefore not return to its origina vaue. Instead, we wi have og z = og r + i2π. (2.13) If starting from this vaue again, describing a compete circuar path aong z = r in the positive direction, the new vaue obtained wi be og z = og r + i4π. (2.14)

25 CHAPTER 2. MATHEMATICAL BACKGROUND 19 The process can be repeated unti a vaue og z = og r + i2πn is obtained. This indicates that the different vaues of og z which are associated with the different vaues of n in equation (2.11) a beong to the same anaytic function. The infinite-vaued anaytic function og z can be decomposed into branches, a of which are singe-vaued, by restricting the vaue of θ to an interva of ength 2π. For instance, by imposing the condition π < θ < π, a branch caed the principa vaue of og z can be obtained. This woud mean that the ogarithmic function cannot cross the negative axis. Crossing the cut wi just be ike going from one branch to the other. Another exampe is the function f(z) = z α, which can aso be written as e α og z. (2.15) The muti-vauedness of z α can be observed by expressing the principa vaue of the ogarithm as og z and writing og z = og z + 2iπn. Then from equation e α og z = e α og z e 2inαπ = P [z α ] e 2iαnπ, (2.16) where P[z α ] is the principa part of the function z α. The vaues of z α are then obtained by mutipying the principa vaue with the factor e 2iαnπ. This shows that z α wi have infinitey many vaues. In particuar, when α is a rationa number of the form m n, with m and n having no common factor and n 1, then the set e 2iαnπ becomes e 2πi( m n )k, (2.17) contains n different numbers. This is obtained by choosing k =, 1, 2,..., n 1 [3]. Geometricay, it is interesting to see how the vaues of z α change when the point z describes a circe about the origin. Recaing the exampe of the ogarithmic function, a given vaue of og z continuousy changes into og z +2πi if z returns to its former position after describing a compete circe about the origin in the positive direction. Accordingy, a given vaue of z α wi aso change into z α e 2iαπ. Repeating the process wi yied z α e 4iαπ, z α e 6iαπ,..., and we see that z α can take any particuar vaue from any other one when z moves around a cosed curve which surrounds the origin in a suitabe way. Just ike in the case of the ogarithmic function, the origin is a singuar point of z α, for the function is not singe-vaued in the neighborhood of z = for a vaues of α.

26 CHAPTER 2. MATHEMATICAL BACKGROUND Riemann surfaces In the study of the ogarithmic function f(z) = og z and of the function f(z) = z α, two mathematica toos were used in order to better understand the nature of many-vaued functions. The first concept was the branchcut, that enabes to singe out one singe-vaued branch of the many-vaued anaytic function. The second was the geometricay suggestive idea of observing how f(z) changes when z starts at a given point and returns to the same point after describing a cosed contour. Both ideas can be combined into a singe method for visuaising the behaviour of a many-vaued function through a geometric construction caed the Riemann surfaces. To iustrate this construction, we consider the simpest case, obtained with the mapping of the function f(z) = z 1 n. As we saw in the preceding section, f(z) wi have n different vaues for any given z (except for z = ). In fact, if we rather consider the equation and if we say [f(z)] n = z, (2.18) z = r(cos w + i sin w), f(z) = ρ(cos φ + i sin φ), (2.19) then from the reation (2.19), we can aso have the equivaences ρ n = r, nφ = w + 2kπ, (2.2) where, ρ = r 1 n which means that r is the n th arithmetic root of the positive number ρ, and φ = ω + 2kπ. (2.21) n To obtain a distinct vaues of f(z), it suffices to give to the arbitrary integer k the n consecutive integra vaues 1, 2,..., n; in this way, we obtain expressions for the n roots of the equation (2.18) as [ ( ) ( )] f(z) = r 1 ω + 2kπ ω + 2kπ n cos + i sin n n = r 1 n e (i ω+2kπ n ) = r 1 iω i2kπ n e n e n = ( re iω) 1 n e i2kπ n = P [ z 1 n ] e i2kπ n (k = 1, 2,..., n), (2.22)

27 CHAPTER 2. MATHEMATICAL BACKGROUND 21 where P [ ] z 1 n is the principa part of f(z). Accordingy, f(z) has n branches. Each of these vaues wi be singe-vaued if z is restricted to the region obtained by cutting the z-pane aong the negative axis [3]. The construction of the Riemann surface ies in the idea that to each of the n branches of f(z), wi be assigned a repica of the cut pane, in which the function is singe-vaued. Indeed, this can aso be understood thinking that there is a one-to-one correspondence between each ange (k 1) 2π n < arg z < k 2π n, k = 1, 2,..., n and f(z), except for the positive axis [31]. This is a mere anaogy of the singe-vaued nature of f(z) in each of the n branches on negative axis. Then the image of each ange (or again of each branch) wi be obtained by performing a cut that wi have an upper and a ower edge, aong the positive axis. Corresponding to the n anges (or branches) in the z-pane, there wi be n identica copies of the f(z)-pane with the cut. These cut-panes are caed the sheets of the Riemann surface, and can be distinguished according to the vaues of z 1 n [ ] by associating the vaue P z 1 n e i2kπ n with the pane of index n. Then these Riemann sheets wi be paced one upon the other in such a way that the (k + 1) th sheet wi be immediatey on top of the k th one, and the corresponding point z in each pane have exacty the same position. If a given point z is now aowed to move aong a cosed curve which surrounds the origin in a positive direction, the path described wi pass from a given branch of the function, say the k th, to another one, say the (k + 1) th one. A geometrica description of the situation is that the upper edge of the cut in the k th pane is attached to the ower edge of the cut in the (k + 1) th pane. The point z = pays a specia roe here. Unike the other points, each of which ies on ony one sheet, the origin connects a the sheets of the surface, and a curve must wind n times around the origin before it coses. A point of this kind, which beongs to more than one sheet of the Riemann surface is caed a branch point Factorization revisited We take a quick break to draw a parae between the overview of Compex Anaysis that has been written up to this point and the process of factorization of the first chapter. In the first chapter, we introduced the Jost functions as the ampitudes of the incoming and outgoing waves in the asymptotics of the radia wave function u (E, r) h ( ) (kr)f (in) (E) + h (+) (kr)f (out) (E). (2.23)

28 CHAPTER 2. MATHEMATICAL BACKGROUND 22 Furthermore, the Ricatti-Hanke functions h (±) (kr) were expicity given as dependent of the momentum k. A few words can be said about the dependence of f (in/out) on k and E = 2 k 2 2µ. Indeed, the momentum can be ( ) expressed as k = 2µ E, and since the energy is compex, it can be put 2 in the exponentia form E = E e iθ, so that k = (2µ 2 ) E e iθ = (2µ 2 ) E e i θ 2, (2.24) ( ) where k = 2µ E wi be the positive square root of k. 2 The impication of the exponentia compex notation of the energy is that the Jost function wi be many-vaued. Indeed, as it was seen in the previous section for the types of function studied, the point E = can be considered as a branching point of the functions f (in/out). If the variabe E describes two fu circes about the origin, the functions f (in/out) wi return to the same vaues as the origina ones. In other words, for any given vaue of the energy on the circe, the momentum k wi have two possibe vaues k = ± (2µ 2 ) E. (2.25) The best way to visuaise the many-vauedness of f (in/out) is to introduce the concept of energetic Riemann surface. As we saw in the case of a function z α with non-integra exponents, if z = 2µ E and α = 1 2 2, here we wi have a function of the type E E defined by E Recaing equation (2.22), we can write [( ) ] 1 2µ 2 2 E = [( ) ] 1 2µ 2 2 E. (2.26) = P { [(2µ 2 ) ] 1 } 2 E e i θ 2 e 2πin 1 2 [ ] E 1 2 e πin (n =, 1), [ ] where P E 1 2 is the principa part of the compex function of the energy. The atter can then be expressed as

29 CHAPTER 2. MATHEMATICAL BACKGROUND 23 E P [ ] E 1 2 e πin (n =, 1). (2.27) The geometric construction of the Riemann surface of the energy is done by considering two parae sheets. When E describes one circe around a branching point, the function of the energy traves on the first sheet, and then continues on the second one unti coming back to the first sheet after competing two circes. In quantum theory, such a continuous transition from one sheet to the other is commony obtained by cutting two exempars of the compex pane aong the positive rea axis and guing the two sheets together in such a way that, if the first sheet is denoted as Γ 1 := and the second sheet as Γ 2 := { [ ] } E C : E P E 1 2 e iα, α < 2π, (2.28) { [ ] } E C : E P E 1 2 e iβ, 2π β < 4π, (2.29) the set (Ω) pictured in Figure (2.3) wi represent a neighborhood of the branching point E R on the Riemann surface. The function of the energy is then singe-vaued on each sheet. Ω E R E R Ω E R iγ (a) first sheet (b) second sheet Figure 2.3: The energetic Riemann surface In the process of factorization discussed in the first chapter, the Jost functions were constructed in such a way that the odd powers of the momentum k were factorised anayticay, eaving the other part dependent ony of even powers of the momentum k, thus making it a singe-vaued function of the energy. From this semi-anaytic expression of the Jost functions, it was then possibe to obtain a system of differentia equations where, very convenienty the factorised part that is responsibe for the existence of branching points was removed, eading to functions in the set of differentia equations that are singe-vaued functions of the energy.

30 CHAPTER 2. MATHEMATICAL BACKGROUND Properties of anaytic functions We briefy discuss some properties of anaytic functions that are of interest to us. Cauchy theorem and Cauchy Integra theorem Most of the genera properties of anaytic functions are embedded in two important theorems: the first one is the Cauchy theorem and the second one is the Cauchy integra theorem. Both wi be given without proof. Cauchy theorem asserts that if f(z) is a singe-vaued anaytic function of z in a region, then C f(z)dz =, (2.3) for any simpe cosed curve C in that region. [28] Cauchy integra theorem states that [3] if f(z) is a singe-vaued anaytic function of z in a region bounded by the simpe cosed curve C, then for any u within C f(u) = 1 2πi C f(z) dz. (2.31) z u The second (Cauchy integra) theorem is of a tremendous importance in the theory of Compex Anaysis as it basicay says that, given a function f(z) that is singe-vaued in a region C and that has a continuous derivative in that region, if the vaues of f within the region are not known but are on the edge of C, then it is possibe to know the vaue of f at some interior point u by simpy cacuating the integra. This means that the vaues of an anaytic function f(z) are competey determined if the vaues on the boundary C are given. Power series expansion An important property of anaytic functions is that they can be expanded in series. In its forma statement, If f(z) is anaytic at z, there exists a Tayor expansion f(z) = a n (z z ) n, vaid in z z < R, (2.32) n= where R is the distance from z to the singuarity nearest z.

31 CHAPTER 2. MATHEMATICAL BACKGROUND 25 Imz R z Rez Figure 2.4: Domain of expansion of the anaytic function f(z) in power series Anaytic continuation aong a path Up to this point, we have seen that anaytic functions are functions differentiabe in a region of the compex pane. However, the properties of power series representations aforementioned can be used to extend the domain of definition of an anaytic function. Let f(z) be an anaytic function in a connected region of the pane, say a circe of radius R and center z. Then f(z) can be defined at a points within the circe by the Tayor series a n (z z ) n. (2.33) Consider a path (γ) starting at z, and a point z 1 on the path and inside the circe of radius R centered at z. Im z z 2 z 1 z (Γ) Re z Figure 2.5: Anaytic continuation of f(z) aong a path If z 1 is not a singuar point of f(z), then the vaues of f(z) are uniquey

32 CHAPTER 2. MATHEMATICAL BACKGROUND 26 determined by the initia conditions at z, and they can themseves be taken as a new set of initia conditions for a new origin at z 1. Accordingy, we can construct a circe with center z 1 and radius R 1. There exists a new Tayor series b n (z z ) n, (2.34) of radius of convergence R 1, whose sum is equa to the sum (2.33) at any point of the domain z z R. The sum (2.34) gives the vaue of f(z) in the circe z z 1 R 1. We repeat the same operation from a point z 2 on the path inside the circe z z 1 R 1, but outside the circe z z R, constructing a circe centered at z 2 and with radius R 2. It foows that a points that can be attained using a ines (Γ) starting from z provided no singuar point is encountered wi form a domain, and that a unique vaue of f(z) at each point z of the domain can be defined. The function f(z) is then anaytic in the domain. This process of finding the vaue of an anaytic function f(z) at a point z, when its vaue is known at the points of some path (Γ) is caed anaytic continuation [29]. 2.2 Existence and nature of soutions of ordinary differentia equations Background Generay, as an introduction to the study of Ordinary Differentia equations of type dy = f(x, y), (2.35) dx exact soutions can be found using eementary methods of integration, such as the method of separation of variabes, or again the method of integrating factors. These types of equations are easiy integrabe on the account that they beong to certain simpe casses. However, it is in genera not evident that a differentia equation of the type of equation (2.35) wi have so eementary a treatment, and very often, the ony recourse is to use methods of numerica approximation. This gives rise to the fundamenta question of the existence of soutions of differentia questions, and interestingy enough, in the chronoogy of the theory of Ordinary Differentia Equations, existence theorems were estabished

33 CHAPTER 2. MATHEMATICAL BACKGROUND 27 ony after the eementary processes of integration aforementioned. Three proofs of these existence theorems are widey found in the itterature. The first one is the cacuus of imits credited to Cauchy. Aso known as the first rigorous investigation to estabish the existence of soutions of a system of ordinary differentia equations, the method of cacuus of imits proves the existence of soutions for anaytic equations through a method of comparison. Cauchy is aso at the origin of another method which does not assume the functions to be anaytic. Athough given by Cauchy and preserved in the ectures of Moigno pubished in 1844 [32], it was greaty simpified by Lipschitz, who gave an expicit account of the necessary hypotheses or the vaidity of the proof. For that reason, the proof is caed the Cauchy-Lipschitz method. The ast of the three existence theorem proofs is the method of successive approximations. This method being the one of interest to us, a description of the theory wi be given The existence theorem Consider the equation dy = f(x, y). (2.36) dx y y + b y y b (x, y ) x a x b M x x + b M x + a x Figure 2.6: Rectanguar domain R surrounding the point (x, y ) Let (x, y ) be a pair of vaues assigned to the rea variabes (x, y) within a rectanguar domain R surrounding the point (x, y ) and defined by the inequaities