Exponentially Accurate Error Estimates Of Quasiclassical Eigenvalues

Exponentially Accurate Error Estimates of Quasiclassical Eigenvalues Julio H. Toloza Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematical Physics Dr. George Hagedorn, Chair Dr. Lay Nam Chang Dr. Martin Klaus Dr. Beate Schmittmann Dr. Werner Kohler December 11, 2002 Blacksburg, Virginia Keywords: Semiclassical Limit, Low-Lying Eigenvalues, Rayleigh-Schr¨odinger Perturbation Theory, Quasimodes Copyright 2002, Julio H. Toloza Exponentially Accurate Error Estimates of Quasiclassical Eigenvalues Julio H. Toloza (ABSTRACT) We study the behavior of truncated Rayleigh-Schr¨odinger series for the low-lying eigenvalues of the time-independent Schr¨odinger equation, in the semiclassical limit ~ & 0. Under certain hypotheses on the potential V (x), we prove that for any given small ~ > 0 there is an optimal truncation of the series for the approximate eigenvalues, such that the difference between an approximate and actual eigenvalue is smaller than exp(−C/~) for some positive constant C. We also prove the analogous results concerning the eigenfunctions. Dedication To my mother. iii Acknowledgments I am greatly indebted to my advisor Dr. George Hagedorn, who always has expressed confidence in my work. His intellectual guidance and support have been crucial to complete this dissertation. I would like to thank Dr. Martin Klaus for his support to enrich and continue my academic career. I am also thankful to my former advisor Dr. Guido Raggio who introduced me to the Mathematical Physics. A very special thanks to Chris Thomas. My life as a graduate student has been much easier because of her ever timely help. Finally, I want to express my gratitude to Natacha, for her patience, support, and unde