Schrödinger Operators: With Applications to Quantum Mechanics and Global Geometry (Google eBook)

Front Cover
Hans L. Cycon
Springer Science & Business Media, 1987 - Computers - 319 pages
1 Review

A complete understanding of Schrödinger operators is a necessary prerequisite for unveiling the physics of nonrelativistic quantum mechanics. Furthermore recent research shows that it also helps to deepen our insight into global differential geometry. This monograph written for both graduate students and researchers summarizes and synthesizes the theory of Schrödinger operators emphasizing the progress made in the last decade by Lieb, Enss, Witten and others. Besides general properties, the book covers, in particular, multiparticle quantum mechanics including bound states of Coulomb systems and scattering theory, quantum mechanics in constant electric and magnetic fields, Schrödinger operators with random and almost periodic potentials and, finally, Schrödinger operator methods in differential geometry to prove the Morse inequalities and the index theorem.

This corrected and extended reprint contains updated proofs and references as well as notes on the development in the field over the past twenty years.

  

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Contents

1 SelfAdjointness
1
12 The Classes S and K
3
13 Katos Inequality and All That
8
14 The LeinfelderSimader Theorem
11
2 L𝑃Properties of Eigenfunctions and All That
13
22 Estimates on Eigenfunctions
17
23 Local Estimates on Gradients
19
24 Eigenfunctions and Spectrum Schnols Theorem
20
74 Howlands Formalism and as Floquet Operators
146
75 Potentials and TimeDependent Problems
149
Notes Added for this Reprint
152
8 Complex Scaling
153
82 Translation Analyticity
158
83 Higher Order Mourre Theory
159
84 Computational Aspects of Complex Scaling
160
85 Complex Scaling and the DCStark Effect
161

25 The AllegrettoPiepenbrink Theorem
22
26 Integral Kernals for exptH
24
3 Geometric Methods for Bound States
27
32 Multiparticle Schrödinger Operators
29
33 The HVZTheorem
32
34 More on the Essential Spectrum
36
Widely Separated Bumps
39
A WarmUp
41
37 The RuskaiSigal Theorem
43
38 Liebs Improvement of the RuskaiSigal Theorem
50
39 NBody Systems with Finitely Many Bound States
52
The StoneWeierstrass Gavotte
58
4 Local Commutator Estimates
60
42 Control of Imbedded Eigenvalues
65
43 Absence of Singular Continuous Spectrum
68
44 Exponential Bounds and Nonexistence
74
45 The Mourre Estimate for NBody Schrödinger Operators
82
5 Phase Space Analysis of Scattering
89
52 Perrys Estimate
92
53 Enss Version of Cooks Method
95
54 RAGE Theorems
97
55 Asymptotics of Observables
101
56 Asymptotic Completeness
105
57 Asymptotic Completeness in the ThreeBody Case
106
Notes Added for this Reprint
114
6 Magnetic Fields
115
61 Gauge Invariance and the Essential Spectrum
117
62 A Schrödinger Operator with Dense Point Spectrum
120
63 Supersymmetry in 0Space Dimensions
121
64 The AharonovCasher Result on Zero Energy Eigenstates
126
65 A Theorem of Iwatsuka
130
66 An Introduction to Other Phenomena in Magnetic Fields
131
Notes Added for this Reprint
134
7 Electric Fields
135
72 A Theorem Needed for the Mourre Theory of the OneDimensional Electric Field
137
73 Propagators for TimeDependent Electric Fields
140
86 Complex Scaling and the ACStark Effect
163
87 Extensions and Generalizations
165
Notes Added for this Reprint
167
9 Random Jacobi Matrices
168
91 Basic Definitions and Results
169
92 The Density of States
175
93 The Lyaponov Exponent and the IshiiPasturKotani Theorem
180
94 Subharmonicity of the Lyaponov Exponent and the Thouless Formula
186
95 Point Spectrum for the Anderson Model
190
Notes Added for this Reprint
201
10 Almost Periodic Jacobi Matrices
203
102 The Almost Mathieu Equation and the Occurrence of Singular Continuous Spectrum
205
103 Pure Point Spectrum and the Maryland Model
209
104 Cantor Sets on Recurrent Absolutely Continuous Spectrum
218
Notes Added for this Reprint
223
11 Wittens Proof of the Morse Inequalities
224
112 The Morse Inequalities
230
113 Hodge Theory
233
114 Wittens Deformed Laplacian
238
115 Proof of Theorem 114
241
12 Patodis Proof of the GaussBonnetChern Theorem and Superproofs of Index Theorems
245
122 The BerezinPatodi Formula
254
Statement and Strategy of the Proof
259
124 Bochner Laplacian and the Weitzenböck Formula
265
125 Elliptic Regularity
272
126 A Canonical Order Calculus
280
127 Cutting and Pasting
290
128 Completion of the Proof of the GaussBonnetChern Theorem
291
129 Mehlers Formula
293
1210 Introduction to the Index Theorem for Dirac Operators
304
Bibliography
307
References Added for this Reprint
321
List of Symbols
325
Subject Index
326
Copyright

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Popular passages

Page 321 - F. Germinet and S. De Bievre. Dynamical localization for discrete and continuous random Schrodinger operators. Comm. Math. Phys., 194(2):323-341, 1998.

About the author (1987)

Barry Simon is IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology.

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