Analysis of Regge Poles in Non-Relativistic Quantum Mechanics Aaron Hiscox UMI Number: U574346
Total Page：16
File Type：pdf, Size：1020Kb
Analysis of Regge poles in non-relativistic quantum mechanics Aaron Hiscox UMI Number: U574346 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U574346 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Bound by BOOKBINDING & PRINTING Unit 3 GabaJfa Workshops Excelsior Ind. Est Cardiff CF14 3AY Tel: (029) 2062 3290 Fax: (029) 2062 5420 For use in the Library only Email: infot9abbeybookbinding.co.uk Web: www.abbeybookbinding.co.uk DECLARATION This work has not previously been accepted in substance for any degree and is not concurrently submitted in candidature for any degree. Signed <4X. .............................(candidate) Date ... .Qrr.l., STATEMENT 1 This thesis is being submitted in partial fulfillment of the requirements for the degree of f u v .(insert MCh, MD, MPhil, PhD etc, as appropriate) Signed ... ........................................... (candidate) Date .... C R ' . j . STATEMENT 2 This thesis is the result of my own independent work/investigation, except where otherwise stated. Other sources are acknowledged by explicit references. S ig n ed .................. (candidate) Date STATEMENT 3 I hereby give consent for my thesis, if accepted, to be available for photocopying and for inter- library loan, and for the title and summary to be made available to outside organisations. Signed A*v iu . Cfo*......................(candidate) Date ... A bstract Regge poles—the name given to poles of the scattering amplitude in the complex angular momentum plane—are of utmost importance in atomic and molecular scattering. We investigate various aspects of non-relativistic Regge pole theory, namely, their behaviour at low energy, cardinality, and sensitivity to boundary conditions. Upon investigation of the former, we find the long-standing conjecture that Regge poles become stable bound states for ultra low energy to be true; the proof is achieved for a potential satisfying the first moment condition at infinity and whose product with the radial variable is bounded near the origin, with the proviso that singular behaviour of the Regge poles may occur. It is known that for an analytic potential V with r2\V(r)\ bounded at the origin and at infinity, there are finitely many Regge poles; we demonstrate that this is still the case for a compactly supported potential which is not as singular as the Coulomb interaction at the origin. This begs the question of whether or not it is possible to explicitly count Regge poles. Not only is this a difficult and interesting mathematical problem, but it also has implications in atomic physics where total cross-sections are often calculated using summations over Regge pole contributions. The author’s attempt at counting Regge poles has revealed an unexpected effect on the Regge poles due to boundary conditions: we show that infinitely many Regge poles go to infinity under nothing more than a change of boundary condition, at least for the free particle case. This work was supported financially by the Engineering and Physical Sciences Research Council (EPSRC) between October 2007 and April 2011. Acknowledgements This thesis would not have been possible without the invaluable contributions of numerous individuals. First and foremost, I would like to deeply and sincerely thank my supervisors Prof. Marco Marietta and Prof. Malcolm Brown. It is absolutely certain that I would not have attained any of these results without their guidance and encouragement—sustained, as it was, over periods during which my progress was difficult to discern. I would especially like to thank Marco for so much of his time during the work on Chapter 4, and I look forward to hopefully continuing this collaboration in the future. I am indebted to Dr. Robert L. Warnock for bringing to my attention an incomplete proof; this has relieved the mind of uncertainty. My research has drawn on a great many published sources, which are recognized at various places throughout the text. However, I must mention the remarkable book on special functions by Prof. Frank W. J. Olver, without which, Chapter 4 would have been many orders of magnitude more difficult; moreover, I would like to thank Prof. Olver for several helpful e-mail correspondences. I am also very grateful to Dr. Karl Michael Schmidt, not only for his help over the last few years, but also for some interesting conversations about physics. Last, but by no means least, I would like to thank all of my family and friends for all their kindness and support. In particular, I thank my wife Jenny for all her love and understanding, and for shielding me from real life when I needed it. Contents 1 Introduction 3 2 R egge Theory 6 2.1 The Radial Schrodinger Equation .......................................................................... 8 2.2 The Scattering Amplitude ....................................................................................... 11 2.3 Partial Wave A n a ly sis ............................................................................................. 13 2.4 The 5 -M atrix ............................................................................................................. 18 2.5 Complex Angular Momentum (CAM) .... ............................................................... 19 2.5.1 Coulomb S c a tte rin g ................................................................................... 22 2.5.2 Regge Representation of the Scattering Amplitude .................................27 2.6 The Wronskian Characterization of Regge Poles .....................................................29 3 Low Energy 30 3.1 Notation and Introduction ..........................................................................................30 3.2 Finite Spherical W ell ............................................................................................... 32 3.2.1 Hankel Function Asymptotics ................................................................... 33 3.3 Compactly Supported Potential ............................................................................ 36 3.4 The General P o te n tia l ............................................................................................ 39 3.4.1 The Zero Energy C a s e ....................................................................................42 3.4.2 Normalizing the Non-Zero Energy Equation ............................................. 44 3.4.3 Existence of the Limiting Wavefunction ............................................... 49 4 Regge Pole Cardinality 52 4.1 Yet Another Characterization of Regge Po les .................................................... 52 4.2 The Integral Equation for the Left-Hand Solution ..................................................53 4.2.1 The Zeros of the Bessel F u n c tio n ............................................................ 53 4.2.2 Two Equivalent K e rn els ............................................................................ 54 4.3 Boundedness of the Normalized K ern el ............................................................... 56 1 CONTENTS 2 4.3.1 The Real Axis ............................................................................................. 56 4.3.2 The Imaginary A x i s ........................................................................................ 63 4.3.3 Growth O rder .....................................................................................................68 4.4 The D’Alembert Reformulation ............................................................................. 71 4.4.1 Langer Asymptotics for the Bessel Function .............................................. 73 5 Boundary Conditions 75 5.1 The Free Problem ................................................................................................... 75 5.1.1 Liouville Normal F o rm ................................................................................. 76 5.1.2 Boundedness of the R esolvent .................................................................... 78 5.1.3 Finding an Eigenvalue Free R eg io n ..............................................................84 6 Conclusion and Future Work 86 A Self-Adjointness 88 A.l Closed Operators ........................................................................................................ 88 A.2 The Adjoint O perator ................................................................................................. 89 A.3 The Sturm-Liouville Operator .....................................................................................91 B The Spectrum 95 B.l Spectral Theory in Finite Dimensions ....................................................................95 B.2 The Resolvent ...............................................................................................................96 B.3 The Essential Spectrum .............................................................................................. 97 B.4 The Glazman Decomposition ...................................................................................101