Mathematical Methods in Quantum Mechanics

1 Mathematical Methodsin Quantum MechanicsWith Applicationsto Schr odinger OperatorsGerald TeschlNote: 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 wantto support the idea of online versions, please consider buying this book: Studiesin MathematicsVolume 99 American Mathematical SocietyProvidence, Rhode IslandEditorial BoardDavid Cox (Chair)Steven G. KrantsRafe MazzeoMartin Scharlemann2000 Mathematics subject , 81 Qxx, 46-01, 34 Bxx, book provides a self-contained introduction to Mathematical Methods in quan-tum Mechanics (spectral theory) with applications to Schr odinger operators.

2 The first part cov-ers Mathematical foundations of Quantum Mechanics from self-adjointness, the spectral theorem, Quantum dynamics (including Stone s and the RAGE theorem) to perturbation theory for self-adjoint second part starts with a detailed study of the free Schr odinger operator respectivelyposition, momentum and angular momentum operators. Then we develop Weyl Titchmarsh the-ory for Sturm Liouville operators and apply it to spherically symmetric problems, in particularto the hydrogen atom. Next we investigate self-adjointness of atomic Schr odinger operators andtheir essential spectrum, in particular the HVZ theorem.

3 Finally we have a look at scatteringtheory and prove asymptotic completeness in the short range additional information and updates on this book, visit: by LATEXand Makeindex. Version: February 12, of Congress Cataloging-in-Publication DataTeschl, Gerald, 1970 Mathematical Methods in Quantum Mechanics : with applications to Schr odinger operators/ Gerald cm. (Graduate Studies in Mathematics ; v. 99)Includes bibliographical references and 978-0-8218-4660-5 (alk. paper)1. Schr odinger operators. 2. Quantum theory Mathematics.

4 I. 20092008045437515 .724 dc22 Copying and readers of this publication, and nonprofit libraries actingfor them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgement of the source is , systematic copying, or multiple reproduction of any material in this pub-lication (including abstracts) is permitted only under license from the American MathematicalSociety.

5 Requests for such permissions should be addressed to the Assistant to the Publisher,American Mathematical Society, Box 6248, Providence, Rhode Island 02940-6248. Requestscan also be made by e-mail 2009 by the American Mathematical Society. All rights American Mathematical Society retains all rightsexcept those granted too the United States Susanne, Simon, and JakobContentsPrefacexiPart 0. PreliminariesChapter 0. A first look at Banach and Hilbert spaces3 Warm up: Metric and topological spaces3 The Banach space of continuous functions12 The geometry of Hilbert spaces16 Completeness22 Bounded operators22 LebesgueLpspaces25 Appendix: The uniform boundedness principle32 Part 1.

6 Mathematical Foundations of Quantum MechanicsChapter 1. Hilbert spaces37 Hilbert spaces37 Orthonormal bases39 The projection theorem and the Riesz lemma43 Orthogonal sums and tensor products45 TheC algebra of bounded linear operators47 Weak and strong convergence49 Appendix: The Stone Weierstra theorem51 Chapter 2. Self-adjointness and spectrum55viiviiiContents Some Quantum mechanics55 Self-adjoint operators58 Quadratic forms and the Friedrichs extension67 Resolvents and spectra73 Orthogonal sums of operators79 Self-adjoint extensions81 Appendix: Absolutely continuous functions84 Chapter 3.

7 The spectral theorem87 The spectral theorem87 More on Borel measures99 Spectral types104 Appendix: The Herglotz theorem106 Chapter 4. Applications of the spectral theorem111 Integral formulas111 Commuting operators115 The min-max theorem117 Estimating eigenspaces119 Tensor products of operators120 Chapter 5. Quantum dynamics123 The time evolution and Stone s theorem123 The RAGE theorem126 The Trotter product formula131 Chapter 6. Perturbation theory for self-adjoint operators133 Relatively bounded operators and the Kato Rellich theorem 133 More on compact operators136 Hilbert Schmidt and trace class operators139 Relatively compact operators and Weyl s theorem145 Relatively form bounded operators and the KLMN theorem 149 Strong and norm resolvent convergence153 Part 2.

8 Schr odinger OperatorsChapter 7. The free Schr odinger operator161 The Fourier transform161 The free Schr odinger operator167 Contentsix The time evolution in the free case169 The resolvent and Green s function171 Chapter 8. Algebraic methods173 Position and momentum173 Angular momentum175 The harmonic oscillator178 Abstract commutation179 Chapter 9. One-dimensional Schr odinger operators181 Sturm Liouville operators181 Weyl s limit circle, limit point alternative187 Spectral transformations I195 Inverse spectral theory202 Absolutely continuous spectrum206 Spectral transformations II209 The spectra of one-dimensional Schr odinger operators214 Chapter 10.

9 One-particle Schr odinger operators221 Self-adjointness and spectrum221 The hydrogen atom222 Angular momentum225 The eigenvalues of the hydrogen atom229 Nondegeneracy of the ground state235 Chapter 11. Atomic Schr odinger operators239 Self-adjointness239 The HVZ theorem242 Chapter 12. Scattering theory247 Abstract theory247 Incoming and outgoing states250 Schr odinger operators with short range potentials253 Part 3. AppendixAppendix A. Almost everything about Lebesgue integration259 Borel measures in a nut shell259 Extending a premeasure to a measure263 Measurable functions268xContents The Lebesgue integral270 Product measures275 Vague convergence of measures278 Decomposition of measures280 Derivatives of measures282 Bibliographical notes289 Bibliography293 Glossary of notation297 Index301 PrefaceOverviewThe present text was written for my courseSchr odinger Operatorsheldat the University of Vienna in winter

10 1999, summer 2002, summer 2005,and winter 2007. It gives a brief but rather self-contained introductionto the Mathematical Methods of Quantum Mechanics with a view towardsapplications to Schr odinger operators. The applications presented are highlyselective and many important and interesting items are not touched 1 is a stripped down introduction to spectral theory of unboundedoperators where I try to introduce only those topics which are needed forthe applications later on. This has the advantage that you will (hopefully)not get drowned in results which are never used again before you get tothe applications.