**AMS Chelsea Publishing**

Volume: 350;
1988;
488 pp;
Hardcover

MSC: Primary 81;
Secondary 03

Print ISBN: 978-0-8218-3624-8

Product Code: CHEL/350.H

List Price: $76.00

AMS Member Price: $68.40

MAA member Price: $68.40

**Electronic ISBN: 978-1-4704-3026-9
Product Code: CHEL/350.H.E**

List Price: $76.00

AMS Member Price: $68.40

MAA member Price: $68.40

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#### Supplemental Materials

# Solvable Models in Quantum Mechanics: Second Edition

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*S. Albeverio; F. Gesztesy; R. Høegh-Krohn; H. Holden*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

This monograph presents a detailed study of a class of
solvable models in quantum mechanics that describe the motion of a
particle in a potential having support at the positions of a discrete
(finite or infinite) set of point sources. Both situations–where
the strengths of the sources and their locations are precisely known
and where these are only known with a given probability
distribution–are covered.

The authors present a systematic mathematical approach to these
models and illustrate its connections with previous heuristic
derivations and computations. Results obtained by different methods in
disparate contexts are thus unified and a systematic control over
approximations to the models, in which the point interactions are
replaced by more regular ones, is provided.

The first edition of this book generated considerable interest for
those learning advanced mathematical topics in quantum mechanics,
especially those connected to the Schrödinger equations. This
second edition includes a new appendix by Pavel Exner, who has
prepared a summary of the progress made in the field since 1988. His
summary, centering around two-body point interaction problems, is
followed by a bibliography focusing on essential developments made
since 1988. appendix by Pavel Exner, who has prepared a summary of
the progress made in the field since 1988. His summary, centering
around two-body point interaction problems, is followed by a
bibliography focusing on essential developments made since 1988.

The material is suitable for graduate students and researchers interested in
quantum mechanics and Schrödinger operators.

#### Readership

Graduate students and research mathematicians interested in quantum mechanics and Schrödinger operators.

#### Reviews & Endorsements

There is a wealth of very pretty examples of Schrödinger operators here which could be presented ... in an elementary quantum mechanics course.

-- MathSciNet

#### Table of Contents

# Table of Contents

## Solvable Models in Quantum Mechanics: Second Edition

- Cover Cover11
- Title page i2
- Quotation iii4
- Preface to the second edition v6
- Preface vii8
- Contents xi12
- Introduction 116
- Part I. The one-center point interaction 924
- Chapter I.1. The one-center point interaction in three dimensions 1126
- Chapter I.2. Coulomb plus one-center point interaction in three dimensions 5267
- Chapter I.3. The one-center 𝛿-interaction in one dimension 7590
- Chapter I.4. The one-center 𝛿’-interaction in one dimension 91106
- Chapter I.5. The one-center point interaction in two dimensions 97112
- Part II. Point interactions with a finite number of centers 107122
- Chapter II.1. Finitely many point interactions in three dimensions 109124
- Chapter II.2. Finitely many 𝛿-interactions in one dimension 140155
- Chapter II.3. Finitely many 𝛿’-interactions in one dimension 154169
- Chapter II.4. Finitely many point interactions in two dimensions 160175
- Part III. Point interactions with infinitely many centers 167182
- Chapter III.1. Infinitely many point interactions in three dimensions 169184
- Chapter III.2. Infinitely many 𝛿-interactions in one dimension 253268
- Chapter III.3. Infinitely many 𝛿’-interactions in one dimension 307322
- Chapter III.4. Infinitely many point interactions in two dimensions 324339
- Chapter III.5. Random Hamiltonians with point interactions 334349
- Appendices 355370
- A. Self-adjoint extensions of symmetric operators 357372
- B. Spectral properties of Hamiltonians defined as quadratic forms 360375
- C. Schrödinger operators with interactions concentrated around infinitely many centers 365380
- D. Boundary conditions for Schrödinger operators on (0,∞) 371386
- E. Time-dependent scattering theory for point interactions 374389
- F. Dirichlet forms for point interactions 376391
- G. Point interactions and scales of Hilbert spaces 380395
- H. Nonstandard analysis and point interactions 386401
- I. Elements of probability theory 396411
- J. Relativistic point interactions in one dimension 399414
- References 413428
- Author Index 441456
- Subject Index 447462
- Appendix K. Seize ans après 453468
- Bibliography 472487
- Errata and addenda 485500
- Back Cover Back Cover1506