Schroedinger operators and mathematical methods in QM

259 Pages·2007·1.166 MB·English

by  Teschl G.

Mathematical Methods
in Quantum Mechanics
With Applications to Schr¨odinger Operators
Gerald Teschl
Gerald Teschl
Fakult¨at fu¨r Mathematik
Nordbergstraße 15
Universita¨t Wien
1090 Wien, Austria
E-mail: [email protected]
URL: http://www.mat.univie.ac.at/˜gerald/
2000 Mathematics subject classification. 81-01, 81Qxx, 46-01
Abstract. This manuscript provides a self-contained introduction to math-
ematicalmethodsinquantummechanics(spectraltheory)withapplications
to Schr¨odinger operators. The first part covers mathematical foundations
of quantum mechanics from self-adjointness, the spectral theorem, quantum
dynamics(includingStone’sandtheRAGEtheorem)toperturbationtheory
for self-adjoint operators.
The second part starts with a detailed study of the free Schr¨odinger op-
erator respectively position, momentum and angular momentum operators.
ThenwedevelopWeyl-TitchmarshtheoryforSturm-Liouvilleoperatorsand
apply it to spherically symmetric problems, in particular to the hydrogen
atom. Next we investigate self-adjointness of atomic Schr¨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.
Keywords and phrases. Schr¨odinger operators, quantum mechanics, un-
bounded operators, spectral theory.
Typeset by AMS-LATEX and Makeindex.
Version: May 10, 2007
Copyright (cid:13)c 1999-2007 by Gerald Teschl

Contents
Preface vii
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 10
§0.3. The geometry of Hilbert spaces 15
§0.4. Completeness 19
§0.5. Bounded operators 20
§0.6. Lebesgue Lp spaces 22
§0.7. Appendix: The uniform boundedness principle 28
Part 1. Mathematical Foundations of Quantum Mechanics
Chapter 1. Hilbert spaces 33
§1.1. Hilbert spaces 33
§1.2. Orthonormal bases 35
§1.3. The projection theorem and the Riesz lemma 39
§1.4. Orthogonal sums and tensor products 40
§1.5. The C∗ algebra of bounded linear operators 42
§1.6. Weak and strong convergence 44
§1.7. Appendix: The Stone–Weierstraß theorem 46
Chapter 2. Self-adjointness and spectrum 49
iii
iv Contents
§2.1. Some quantum mechanics 49
§2.2. Self-adjoint operators 52
§2.3. Resolvents and spectra 64
§2.4. Orthogonal sums of operators 69
§2.5. Self-adjoint extensions 71
§2.6. Appendix: Absolutely continuous functions 74
Chapter 3. The spectral theorem 77
§3.1. The spectral theorem 77
§3.2. More on Borel measures 88
§3.3. Spectral types 91
§3.4. Appendix: The Herglotz theorem 93
Chapter 4. Applications of the spectral theorem 99
§4.1. Integral formulas 99
§4.2. Commuting operators 102
§4.3. The min-max theorem 105
§4.4. Estimating eigenspaces 106
§4.5. Tensor products of operators 107
Chapter 5. Quantum dynamics 109
§5.1. The time evolution and Stone’s theorem 109
§5.2. The RAGE theorem 112
§5.3. The Trotter product formula 117
Chapter 6. Perturbation theory for self-adjoint operators 119
§6.1. Relatively bounded operators and the Kato–Rellich theorem 119
§6.2. More on compact operators 121
§6.3. Hilbert–Schmidt and trace class operators 124
§6.4. Relatively compact operators and Weyl’s theorem 130
§6.5. Strong and norm resolvent convergence 134
Part 2. Schr¨odinger Operators
Chapter 7. The free Schr¨odinger operator 141
§7.1. The Fourier transform 141
§7.2. The free Schr¨odinger operator 146
§7.3. The time evolution in the free case 148
§7.4. The resolvent and Green’s function 149
Contents v
Chapter 8. Algebraic methods 153
§8.1. Position and momentum 153
§8.2. Angular momentum 155
§8.3. The harmonic oscillator 158
Chapter 9. One dimensional Schr¨odinger operators 161
§9.1. Sturm-Liouville operators 161
§9.2. Weyl’s limit circle, limit point alternative 165
§9.3. Spectral transformations 172
Chapter 10. One-particle Schr¨odinger operators 181
§10.1. Self-adjointness and spectrum 181
§10.2. The hydrogen atom 182
§10.3. Angular momentum 185
§10.4. The eigenvalues of the hydrogen atom 188
§10.5. Nondegeneracy of the ground state 190
Chapter 11. Atomic Schr¨odinger operators 193
§11.1. Self-adjointness 193
§11.2. The HVZ theorem 195
Chapter 12. Scattering theory 201
§12.1. Abstract theory 201
§12.2. Incoming and outgoing states 204
§12.3. Schr¨odinger operators with short range potentials 206
Part 3. Appendix
Appendix A. Almost everything about Lebesgue integration 213
§A.1. Borel measures in a nut shell 213
§A.2. Extending a premasure to a measure 217
§A.3. Measurable functions 222
§A.4. The Lebesgue integral 224
§A.5. Product measures 229
§A.6. Decomposition of measures 231
§A.7. Derivatives of measures 234
Bibliography 239
Glossary of notations 241
Index 245

Preface
Overview
ThepresentmanuscriptwaswrittenformycourseSchr¨odingerOperators
heldattheUniversityofViennainWinter1999,Summer2002,andSummer
2005. It is supposed to give a brief but rather self contained introduction
to the mathematical methods of quantum mechanics with a view towards
applicationstoSchr¨odingeroperators. Theapplicationspresentedarehighly
selective and many important and interesting items are not touched.
The first part is a stripped down introduction to spectral theory of un-
bounded operators where I try to introduce only those topics which are
needed for the applications later on. This has the advantage that you will
not get drowned in results which are never used again before you get to
the applications. In particular, I am not trying to provide an encyclope-
dic reference. Nevertheless I still feel that the first part should give you a
solid background covering all 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,existenceofspectralmeasuresisestablishedviatheHerglotzrather
than the Riesz representation theorem since this approach paves the way for
aninvestigationofspectraltypesviaboundaryvaluesoftheresolventasthe
spectral parameter approaches the real line.
vii
viii Preface
The second part starts with the free Schr¨odinger equation and computes
the free resolvent and time evolution. In addition, I discuss position, mo-
mentum, and angular momentum operators via algebraic methods. This is
usually found in any physics textbook on quantum mechanics, with the only
difference that I include some technical details which are usually not found
there. Furthermore, I compute the spectrum of the hydrogen atom, 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.
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.
Readers guide
There is some intentional overlap between Chapter 0, Chapter 1 and
Chapter 2. Hence, provided you have the necessary background, you can
startreadinginChapter1orevenChapter2. Chapters2, 3arekeychapters
and you should study them in detail (except for Section 2.5 which can be
skipped on first reading). Chapter 4 should give you an idea of how the
spectral theorem is used. You should have a look at (e.g.) the first section
andyoucancomebacktotheremainingonesasneeded. Chapter5contains
two key results from quantum dynamics, Stone’s theorem and the RAGE
theorem. In particular the RAGE theorem shows the connections between
long time behavior and spectral types. Finally, Chapter 6 is again of central
importance and should be studied in detail.
The chapters in the second part are mostly independent of each others
except for the first one, Chapter 7, which is a prerequisite for all others
except for Chapter 9.
If you are interested in one dimensional models (Sturm-Liouville equa-
tions), Chapter 9 is all you need.
If you are interested in atoms, read Chapter 7, Chapter 10, and Chap-
ter 11. In particular, you can skip the separation of variables (Sections 10.3
Preface ix
and 10.4, which require Chapter 9) method for computing the eigenvalues of
the Hydrogen atom if you are happy with the fact that there are countably
many which accumulate at the bottom of the continuous spectrum.
If you are interested in scattering theory, read Chapter 7, the first two
sections of Chapter 10, and Chapter 12. Chapter 5 is one of the key prereq-
uisites in this case.
Availability
It is available from
http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/
Acknowledgments
I’d like to thank Volker Enß for making his lecture notes available to
me and Maria Hoffmann-Ostenhof, Zhenyou Huang, Helge Kru¨ger, Wang
Lanning, Arnold L. Neidhardt, Harald Rindler, and Karl Unterkofler for
pointing out errors in previous versions.
Gerald Teschl
Vienna, Austria
February, 2005