Solvable Models in Quantum Mechanics With Appendix Written By Pavel Exner, Second Edition (AMS Chelsea Publishing)

✍ Scribed by Pavel Exner, Sergio Albeverio

• Publisher: Amer Mathematical Society
• Year: 2004
• Tongue: English
• Leaves: 505
• Edition: second
• Category: Library

✦ Synopsis

The 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 monograph generated considerable interest for those learning advanced mathematical topics in quantum mechanics, especially those connected to the Schr?dinger 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. The material is suitable for graduate students and researchers interested in quantum mechanics and Schr?dinger operators.

✦ Table of Contents

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface to second edition......Page 6
Preface......Page 8
Contents......Page 12
Introduction......Page 16
PART I The One-Center Point Interaction......Page 24
I.1.1 Basic Properties......Page 26
I.1.2 Approximations by Means of Local as well as Nonlocal Scaled Short-Range Interactions......Page 32
I.1.3 Convergence of Eigenvalues and Resonances......Page 43
I.1.4 Stationary Scattering Theory......Page 52
Notes......Page 61
I.2.1 Basic Properties......Page 67
I.2.2 Approximations by Means of Scaled Coulomb-Type Interactions......Page 72
I.2.3 Stationary Scattering Theory......Page 81
Notes......Page 89
I.3.1 Basic Properties......Page 90
I.3.2 Approximations by Means of Local Scaled Short-Range Interactions......Page 94
I.3.3 Convergence of Eigenvalues and Resonances......Page 98
I.3.4 Stationary Scattering Theory......Page 100
Notes......Page 104
CHAPTER I.4 The One-Center b'-interaction in One Dimension......Page 106
Notes......Page 110
CHAPTER I.5 The One-Center Point Interaction in Two Dimensions......Page 112
Notes......Page 120
PART II Point Interactions with a Finite Number of Centers......Page 122
II.1.1 Basic Properties......Page 124
II.1.2 Approximations by Means of Local Scaled Short-Range Interactions......Page 136
II.1.3 Convergence of Eigenvalues and Resonances......Page 140
II.1.4 Multiple Well Problems......Page 147
II.1.5 Stationary Scattering Theory......Page 149
Notes......Page 153
II.2.1 Basic Properties......Page 155
II.2.2 Approximations by Means of Local Scaled Short-Range Interactions......Page 160
II.2.3 Convergence of Eigenvalues and Resonances......Page 163
II.2.4 Stationary Scattering Theory......Page 165
Notes......Page 168
CHAPTER II.3 Finitely Many 8'-Interactions in One Dimension......Page 169
Notes......Page 174
CHAPTER II.4 Finitely Many Point Interactions in Two Dimensions......Page 175
Notes......Page 180
PART III Point Interactions with Infinitely Many Centers......Page 182
III.1.1 Basic Properties......Page 184
III.1.2 Approximations by Means of Local Scaled Short-Range Interactions......Page 188
III.1.3 Periodic Point Interactions......Page 191
III.1.4 Crystals......Page 193
III.1.5 Straight Polymers......Page 215
III.1.6 Monomolecular Layers......Page 225
III.1.7 Bragg Scattering......Page 232
III.1.8 Fermi Surfaces......Page 241
III.1.9 Crystals with Defects and Impurities......Page 254
Notes......Page 265
III.2.1 Basic Properties......Page 268
III.2.2 Approximations by Means of Local Scaled Short-Range Interactions......Page 276
III.2.3 Periodic 6-Interactions......Page 278
III.2.4 Half-Crystals......Page 299
III.2.5 Quasi-periodic b-Interactions......Page 303
III.2.6 Crystals with Defects and Impurity Scattering......Page 305
Notes......Page 318
CHAPTER III.3 Infinitely Many b'-Interactions in One Dimension......Page 322
Notes......Page 338
CHAPTER III.4 Infinitely Many Point Interactions in Two Dimensions......Page 339
Notes......Page 348
III.5.1 Preliminaries......Page 349
III.5.2 Random Point Interactions in Three Dimensions......Page 356
III.5.3 Random Point Interactions in One Dimension......Page 364
Notes......Page 368
APPENDICES......Page 370
A Self-Adjoint Extensions of Symmetric Operators......Page 372
B Spectral Properties of Hamiltonians Defined as Quadratic Forms......Page 375
C Schrodinger Operators with Interactions Concentrated Around Infinitely Many Centers......Page 380
D Boundary Conditions for Schrodinger Operators on (0, \infty)......Page 386
E Time-Dependent Scattering Theory for Point Interactions......Page 389
F Dirichlet Forms for Point Interactions......Page 391
G Point Interactions and Scales of Hilbert Spaces......Page 395
H.1 A Very Short Introduction to Nonstandard Analysis......Page 401
H.2 Point Interactions Using Nonstandard Analysis......Page 406
I Elements of Probability Theory......Page 411
J Relativistic Point Interactions in One Dimension......Page 414
K Seize ans apres......Page 468
References......Page 428
Index......Page 456
Bibliography......Page 487
Errata and Addenda......Page 500

✦ Subjects

Физика;Общая и теоретическая физика;