Mathematical methods in quantum mechanics gerald teschl

www.elsolucionario.net Mathematical Methods in Quantum Mechanics With Applications to Schrăodinger Operators Gerald Teschl Note: The AMS has granted the permission to post this online edition! This version is for personal online use only! If you like this book and want to support the idea of online versions, please consider buying this book: http://www.ams.org/bookstore-getitem?item=gsm-99 Graduate Studies in Mathematics Volume 99 American Mathematical Society Providence, Rhode Island www.elsolucionario.net Editorial Board David Cox (Chair) Steven G Krants Rafe Mazzeo Martin Scharlemann 2000 Mathematics subject classification 81-01, 81Qxx, 46-01, 34Bxx, 47B25 Abstract This book provides a self-contained introduction to mathematical methods in quantum mechanics (spectral theory) with applications to Schră odinger operators The first part covers mathematical foundations of quantum mechanics from self-adjointness, the spectral theorem, quantum dynamics (including Stone’s and the RAGE theorem) to perturbation theory for selfadjoint operators The second part starts with a detailed study of the free Schră odinger operator respectively position, momentum and angular momentum operators Then we develop Weyl–Titchmarsh theory for Sturm–Liouville operators and apply it to spherically symmetric problems, in particular to the hydrogen atom Next we investigate self-adjointness of atomic Schră odinger operators and their essential spectrum, in particular the HVZ theorem Finally we have a look at scattering theory and prove asymptotic completeness in the short range case For additional information and updates on this book, visit: http://www.ams.org/bookpages/gsm-99/ Typeset by LATEXand Makeindex Version: February 17, 2009 Library of Congress Cataloging-in-Publication Data Teschl, Gerald, 1970– Mathematical methods in quantum mechanics : with applications to Schră odinger operators / Gerald Teschl p cm — (Graduate Studies in Mathematics ; v 99) Includes bibliographical references and index ISBN 978-0-8218-4660-5 (alk paper) Schră odinger operators Quantum theoryMathematics I Title QC174.17.S3T47 2009 2008045437 515’.724–dc22 Copying and reprinting Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society Requests for such permissions should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O Box 6248, Providence, Rhode Island 02940-6248 Requests can also be made by e-mail to reprint-permission@ams.org c 2009 by the American Mathematical Society All rights reserved The American Mathematical Society retains all rights except those granted too the United States Government www.elsolucionario.net To Susanne, Simon, and Jakob www.elsolucionario.net www.elsolucionario.net Contents Preface xi Part Preliminaries Chapter A first look at Banach and Hilbert spaces §0.1 Warm up: Metric and topological spaces §0.2 The Banach space of continuous functions 12 §0.3 The geometry of Hilbert spaces 16 §0.4 Completeness 22 §0.5 Bounded operators 22 Lp §0.6 Lebesgue §0.7 Appendix: The uniform boundedness principle spaces 25 32 Part Mathematical Foundations of Quantum Mechanics Chapter Hilbert spaces 37 §1.1 Hilbert spaces 37 §1.2 Orthonormal bases 39 §1.3 The projection theorem and the Riesz lemma 43 §1.4 Orthogonal sums and tensor products 45 C∗ §1.5 The §1.6 Weak and strong convergence 49 §1.7 Appendix: The Stone–Weierstraß theorem 51 Chapter algebra of bounded linear operators Self-adjointness and spectrum 47 55 vii www.elsolucionario.net viii §2.1 §2.2 §2.3 §2.4 §2.5 §2.6 §2.7 Contents Some quantum mechanics Self-adjoint operators Quadratic forms and the Friedrichs extension Resolvents and spectra Orthogonal sums of operators Self-adjoint extensions Appendix: Absolutely continuous functions 55 58 67 73 79 81 84 Chapter §3.1 §3.2 §3.3 §3.4 The spectral theorem The spectral theorem More on Borel measures Spectral types Appendix: The Herglotz theorem 87 87 99 104 107 Chapter §4.1 §4.2 §4.3 §4.4 §4.5 Applications of the spectral theorem Integral formulas Commuting operators The min-max theorem Estimating eigenspaces Tensor products of operators 111 111 115 117 119 120 Chapter §5.1 §5.2 §5.3 Quantum dynamics The time evolution and Stone’s theorem The RAGE theorem The Trotter product formula 123 123 126 131 Chapter §6.1 §6.2 §6.3 §6.4 §6.5 §6.6 Perturbation theory for self-adjoint operators Relatively bounded operators and the Kato–Rellich theorem More on compact operators Hilbert–Schmidt and trace class operators Relatively compact operators and Weyl’s theorem Relatively form bounded operators and the KLMN theorem Strong and norm resolvent convergence 133 133 136 139 145 149 153 Part Schră odinger Operators Chapter The free Schră odinger operator Đ7.1 The Fourier transform Đ7.2 The free Schră odinger operator 161 161 167 www.elsolucionario.net Contents §7.3 §7.4 ix The time evolution in the free case The resolvent and Green’s function 169 171 Chapter §8.1 §8.2 §8.3 §8.4 Algebraic methods Position and momentum Angular momentum The harmonic oscillator Abstract commutation 173 173 175 178 179 Chapter §9.1 §9.2 §9.3 §9.4 §9.5 §9.6 §9.7 One-dimensional Schrăodinger operators SturmLiouville operators Weyls limit circle, limit point alternative Spectral transformations I Inverse spectral theory Absolutely continuous spectrum Spectral transformations II The spectra of one-dimensional Schrăodinger operators 181 181 187 195 202 206 209 214 Chapter 10 One-particle Schrăodinger operators §10.1 Self-adjointness and spectrum §10.2 The hydrogen atom §10.3 Angular momentum §10.4 The eigenvalues of the hydrogen atom §10.5 Nondegeneracy of the ground state 221 221 222 225 229 235 Chapter 11 Atomic Schră odinger operators Đ11.1 Self-adjointness Đ11.2 The HVZ theorem 239 239 242 Chapter 12 Scattering theory Đ12.1 Abstract theory Đ12.2 Incoming and outgoing states Đ12.3 Schră odinger operators with short range potentials 247 247 250 253 Part Appendix Appendix A Almost everything about Lebesgue integration §A.1 Borel measures in a nut shell §A.2 Extending a premeasure to a measure §A.3 Measurable functions 259 259 263 268 www.elsolucionario.net x Contents §A.4 The Lebesgue integral 270 §A.5 Product measures 275 §A.6 Vague convergence of measures 278 §A.7 Decomposition of measures 280 §A.8 Derivatives of measures 282 Bibliographical notes 289 Bibliography 293 Glossary of notation 297 Index 301 www.elsolucionario.net Preface Overview The present text was written for my course Schră odinger Operators held at the University of Vienna in winter 1999, summer 2002, summer 2005, and winter 2007 It gives a brief but rather self-contained introduction to the mathematical methods of quantum mechanics with a view towards applications to Schră odinger operators The applications presented are highly selective and many important and interesting items are not touched upon Part is a stripped down introduction to spectral theory of unbounded operators where I try to introduce only those topics which are needed for the applications later on This has the advantage that you will (hopefully) not get drowned in results which are never used again before you get to the applications In particular, I am not trying to present an encyclopedic reference Nevertheless I still feel that the first part should provide a solid background covering many important results which are usually taken for granted in more advanced books and research papers My approach is built around the spectral theorem as the central object Hence I try to get to it as quickly as possible Moreover, I not take the detour over bounded operators but I go straight for the unbounded case In addition, existence of spectral measures is established via the Herglotz theorem rather than the Riesz representation theorem since this approach paves the way for an investigation of spectral types via boundary values of the resolvent as the spectral parameter approaches the real line xi ... Cataloging -in- Publication Data Teschl, Gerald, 1970– Mathematical methods in quantum mechanics : with applications to Schră odinger operators / Gerald Teschl p cm — (Graduate Studies in Mathematics... discontinuous (Problem 0.18)! This shows that in infinite dimensional spaces different norms will give rise to different convergent sequences! In fact, the key to solving problems in infinite... Martin Scharlemann 2000 Mathematics subject classification 81-01, 81Qxx, 46-01, 34Bxx, 47B25 Abstract This book provides a self-contained introduction to mathematical methods in quantum mechanics