Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators SECOND EDITION Gerald Teschl Graduate Studies in Mathematics Volume 157 American Mathematical Society Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators, Second Edition https://doi.org/10.1090//gsm/157 Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators, Second Edition Gerald Teschl Graduate Studies in Mathematics Volume 157 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 81-01, 81Qxx, 46-01, 34Bxx, 47B25. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-157 Library of Congress Cataloging-in-Publication Data Teschl, Gerald, 1970– Mathematical methods in quantum mechanics : with applications to Schr¨odinger operators / Gerald Teschl.– Second edition. pages cm. — (Graduate studies in mathematics ; volume 157) Includes bibliographical references and index. ISBN 978-1-4704-1704-8 (alk. paper) 1. Schr¨odinger operator. 2. Quantum theory—Mathematics. I. Title. QC174.17.S3T47 2014 530.120151—dc23 2014019123 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 select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. 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Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 To Susanne, Simon, and Jakob Contents Preface xi Part 0. Preliminaries Chapter 0. A first look at Banach and Hilbert spaces 3 §0.1. Warm up: Metric and topological spaces 3 §0.2. The Banach space of continuous functions 14 §0.3. The geometry of Hilbert spaces 21 §0.4. Completeness 26 §0.5. Bounded operators 27 §0.6. Lebesgue Lp spaces 30 §0.7. Appendix: The uniform boundedness principle 38 Part 1. Mathematical Foundations of Quantum Mechanics Chapter 1. Hilbert spaces 43 §1.1. Hilbert spaces 43 §1.2. Orthonormal bases 45 §1.3. The projection theorem and the Riesz lemma 49 §1.4. Orthogonal sums and tensor products 52 §1.5. The C∗ algebra of bounded linear operators 54 §1.6. Weak and strong convergence 55 §1.7. Appendix: The Stone–Weierstraß theorem 59 Chapter 2. Self-adjointness and spectrum 63 vii viii Contents §2.1. Some quantum mechanics 63 §2.2. Self-adjoint operators 66 §2.3. Quadratic forms and the Friedrichs extension 76 §2.4. Resolvents and spectra 83 §2.5. Orthogonal sums of operators 89 §2.6. Self-adjoint extensions 91 §2.7. Appendix: Absolutely continuous functions 95 Chapter 3. The spectral theorem 99 §3.1. The spectral theorem 99 §3.2. More on Borel measures 112 §3.3. Spectral types 118 §3.4. Appendix: Herglotz–Nevanlinna functions 120 Chapter 4. Applications of the spectral theorem 131 §4.1. Integral formulas 131 §4.2. Commuting operators 135 §4.3. Polar decomposition 138 §4.4. The min-max theorem 140 §4.5. Estimating eigenspaces 142 §4.6. Tensor products of operators 143 Chapter 5. Quantum dynamics 145 §5.1. The time evolution and Stone’s theorem 145 §5.2. The RAGE theorem 150 §5.3. The Trotter product formula 155 Chapter 6. Perturbation theory for self-adjoint operators 157 §6.1. Relatively bounded operators and the Kato–Rellich theorem 157 §6.2. More on compact operators 160 §6.3. Hilbert–Schmidt and trace class operators 163 §6.4. Relatively compact operators and Weyl’s theorem 170 §6.5. Relatively form-bounded operators and the KLMN theorem 174 §6.6. Strong and norm resolvent convergence 179 Part 2. Schr¨odinger Operators Chapter 7. The free Schr¨odinger operator 187 §7.1. The Fourier transform 187 Contents ix §7.2. Sobolev spaces 194 §7.3. The free Schr¨odinger operator 197 §7.4. The time evolution in the free case 199 §7.5. The resolvent and Green’s function 201 Chapter 8. Algebraic methods 207 §8.1. Position and momentum 207 §8.2. Angular momentum 209 §8.3. The harmonic oscillator 212 §8.4. Abstract commutation 214 Chapter 9. One-dimensional Schr¨odinger operators 217 §9.1. Sturm–Liouville operators 217 §9.2. Weyl’s limit circle, limit point alternative 223 §9.3. Spectral transformations I 231 §9.4. Inverse spectral theory 238 §9.5. Absolutely continuous spectrum 242 §9.6. Spectral transformations II 245 §9.7. The spectra of one-dimensional Schr¨odinger operators 250 Chapter 10. One-particle Schr¨odinger operators 257 §10.1. Self-adjointness and spectrum 257 §10.2. The hydrogen atom 258 §10.3. Angular momentum 261 §10.4. The eigenvalues of the hydrogen atom 265 §10.5. Nondegeneracy of the ground state 272 Chapter 11. Atomic Schr¨odinger operators 275 §11.1. Self-adjointness 275 §11.2. The HVZ theorem 278 Chapter 12. Scattering theory 283 §12.1. Abstract theory 283 §12.2. Incoming and outgoing states 286 §12.3. Schr¨odinger operators with short range potentials 289 Part 3. Appendix Appendix A. Almost everything about Lebesgue integration 295 §A.1. Borel measures in a nutshell 295 x Contents §A.2. Extending a premeasure to a measure 303 §A.3. Measurable functions 307 §A.4. How wild are measurable objects? 309 §A.5. Integration — Sum me up, Henri 312 §A.6. Product measures 319 §A.7. Transformation of measures and integrals 322 §A.8. Vague convergence of measures 328 §A.9. Decomposition of measures 331 §A.10. Derivatives of measures 334 Bibliographical notes 341 Bibliography 345 Glossary of notation 349 Index 353 Preface Overview The present text was written for my course Schr¨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 Schr¨odinger operators. The applications presented are highly selective; as a result, many important and interesting items are not touched upon. Part 1 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 do 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 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 xii Preface Part 2 starts with the free Schr¨odinger equation and computes the free resolvent and time evolution. In addition, I discuss position, momentum, and angular momentum operators via algebraic methods. This is usu- ally found in any physics textbook on quantum mechanics, with the only difference being that I include some technical details which are typically not found there. Then there is an introduction to one-dimensional mod- els (Sturm–Liouville operators) including generalized eigenfunction expan- sions (Weyl–Titchmarsh theory) and subordinacy theory from Gilbert and Pearson. These results are applied to compute the spectrum of the hy- drogen atom, where again I try to provide some mathematical details not found in physics textbooks. Further topics are nondegeneracy of the ground state, spectra of atoms (the HVZ theorem), and scattering theory (the Enß method). Prerequisites I assume some previous experience with Hilbert spaces and bounded linear operators which should be covered in any basic course on functional analysis. However, while this assumption is reasonable for mathematics students, it might not always be for physics students. For this reason there is a preliminary chapter reviewing all necessary results (including proofs). In addition, there is an appendix (again with proofs) providing all necessary results from measure theory. Literature The present book is highly influenced by the four volumes of Reed and Simon [49]–[52](seealso[16]) and by the book by Weidmann [70](anex- tended version of which has recently appeared in two volumes [72], [73], however, only in German). Other books with a similar scope are, for ex- ample, [16], [17], [21], [26], [28], [30], [48], [57], [63], and [65].