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Schrodinger Operators : With Applications to Quantum Mechanics and Global Geometry - H. Cycon

Schrodinger Operators

With Applications to Quantum Mechanics and Global Geometry

By: H. Cycon


Published: 31st December 2007
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Are you looking for a concise summary of the theory of Schrodinger operators? Here it is. Emphasizing the progress made in the last decade by Lieb, Enss, Witten and others, the three authors don't just cover general properties, but also detail multiparticle quantum mechanics - including bound states of Coulomb systems and scattering theory. This corrected and extended reprint contains updated references as well as notes on the development in the field over the past twenty years.

Self-Adjointnessp. 1
Basic Perturbation Theoremsp. 1
The Classes S? and K?p. 3
Kato's Inequality and All Thatp. 8
The Leinfelder-Simader Theoremp. 11
Lp-Properties of Eigenfunctions, and All Thatp. 13
Semigroup Propertiesp. 13
Estimates on Eigenfunctionsp. 17
Local Estimates on Gradientsp. 19
Eigenfunctions and Spectrum (Sch'nol's Theorem)p. 20
The Allegretto-Piepenbrink Theoremp. 22
Integral Kernels for exp(-tH)p. 24
Geometric Methods for Bound Statesp. 27
Partitions of Unity and the IMS Localization Formulap. 27
Multiparticle Schrödinger Operatorsp. 29
The HVZ-Theoremp. 32
More on the Essential Spectrump. 36
A Theorem of Klaus: Widely Separated Bumpsp. 39
Applications to Atomic Physics: A Warm-upp. 41
The Ruskai-Sigal Theoremp. 43
Lieb's Improvement of the Ruskai-Sigal Theoremp. 50
N-Body Systems with Finitely Many Bound Statesp. 52
Appendix: The Stone-Weierstrass Gavottep. 58
Local Commutator Estimatesp. 60
Putnam's Theorem and the Mourre Estimatep. 60
Control of Imbedded Eigenvaluesp. 65
Absence of Singular Continuous Spectrump. 67
Exponential Bounds and Nonexistence of Positive Eigenvaluesp. 74
The Mourre Estimate for N-Body Schrodinger Operatorsp. 82
Phase Space Analysis of Scatteringp. 89
Some Notions of Scattering Theoryp. 89
Perry's Estimatep. 92
Enss' Version of Cook's Methodp. 95
RAGE Theoremsp. 97
Asymptotics of Observablesp. 101
Asymptotic Completenessp. 105
Asymptotic Completeness in the Three-Body Casep. 106
Notes Added for this Reprintp. 114
Magnetic Fieldsp. 115
Gauge Invariance and the Essential Spectrump. 117
A Schrodinger Operator with Dense Point Spectrump. 120
Supersymmetry (in 0-Space Dimensions)p. 121
The Aharonov-Casher Result on Zero Energy Eigenstatesp. 126
Theorem of Iwatsukap. 130
An Introduction to Other Phenomena in Magnetic Fieldsp. 131
Notes Added for this Reprintp. 134
Electric Fieldsp. 135
The Two-Body Stark Effectp. 135
A Theorem Needed for the Mourre Theory of the One-Dimensional Electric Fieldp. 137
Propagators for Time-Dependent Electric Fieldsp. 140
Howland's Formalism and Floquet Operatorsp. 146
Potentials and Time-Dependent Problemsp. 149
Notes Added for this Reprintp. 152
Complex Scalingp. 153
Review of "Ordinary" Complex Scalingp. 153
Translation Analyticityp. 158
Higher Order Mourre Theoryp. 159
Computational Aspects of Complex Scalingp. 160
Complex Scaling and the DC-Stark Effectp. 161
Complex Scaling and the AC-Stark Effectp. 163
Extensions and Generalizationsp. 165
Notes Added for this Reprintp. 167
Random Jacobi Matricesp. 168
Basic Definitions and Resultsp. 169
The Density of Statesp. 175
The Lyaponov Exponent and the Ishii-Pastur-Kotani Theoremp. 180
Subharmonicity of the Lyaponov Exponent and the Thouless Formulap. 186
Point Spectrum for the Anderson Modelp. 190
Notes Added for this Reprintp. 201
Almost Periodic Jacobi Matricesp. 203
Almost Periodic Sequences and Some General Resultsp. 203
The Almost Mathieu Equation and the Occurrence of Singular Continuous Spectrump. 205
Pure Point Spectrum and the Maryland Modelp. 209
Cantor Sets and Recurrent Absolutely Continuous Spectrump. 218
Notes Added for this Reprintp. 223
Witten's Proof of the Morse Inequalitiesp. 224
The Quasiclassical Eigenvalue Limitp. 224
The Morse Inequalitiesp. 230
Hodge Theoryp. 233
Witten's Deformed Laplacianp. 238
Proof of Theorem 11.4p. 241
Patodi's Proof of the Gauss-Bonnet-Chern Theorem and Superproofs of Index Theoremsp. 245
A Very Rapid Course in Riemannian Geometryp. 245
The Berezin-Patodi Formulap. 254
The Gauss-Bonnet-Chern Theorem: Statement and Strategy of the Proofp. 259
Bochner Laplacian and the Weitzenbock Formulap. 265
Elliptic Regularityp. 272
A Canonical Order Calculusp. 280
Cutting and Pastingp. 290
Completion of the Proof of the Gauss-Bonnet-Chern Theoremp. 291
Mehler's Formulap. 291
Introduction to the Index Theorem for Dirac Operatorsp. 304
Bibliographyp. 307
References Added for this Reprintp. 321
List of Symbolsp. 325
Subject Indexp. 327
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540167587
ISBN-10: 3540167587
Series: Theoretical and Mathematical Physics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 329
Published: 31st December 2007
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.83
Weight (kg): 0.48