Scattering theory of classical and quantum N-particle systems

Jan Derezinski, Christian Gerard9783642082849, 364208284X

This monograph addresses researchers and students. It is a modern presentation of time-dependent methods for studying problems of scattering theory in classical and quantum mechanics. Particular attention is paid to long-range potentials. For a large class of interactions the existence of the asymptotic velocity and the asymptotic completeness of the wave operators is shown. The book is self-contained and explains in detail concepts that deepen the understanding. A special feature is its emphasis to show the beautiful analogy between classical and quantum scattering theory.

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Table of contents :
Cover ……Page 1
Dedication ……Page 2
Acknowledgments ……Page 3
0 Introduction ……Page 4
1.0 Introduction ……Page 10
1.1 Basic Notation ……Page 16
1.2 Newton’s Equation ……Page 18
1.3 Asymptotic Momentum ……Page 19
1.4 Fast-Decaying Case ……Page 20
1.5 Slow-Decaying Case I ……Page 25
1.6 Slow-Decaying Case II ……Page 31
1.7 Boundary Conditions for Wave Transformations ……Page 36
1.8 Conservative Forces ……Page 38
1.9 Gauge Invariance of Wave Transformations ……Page 41
1.10 Smoothness of Trajectories ……Page 51
1.11 Comparison of Two Dynamics ……Page 55
1.12 More Examples of Modified Free Dynamics ……Page 58
2.0 Introduction ……Page 62
2.1 General Facts about Dynamical Systems ……Page 65
2.2 Upper Bounds on Trajectories ……Page 67
2.3 The Mourre Estimate and Scattering Trajectories ……Page 70
2.4 Non-Trapping Energies ……Page 74
2.5 Asymptotic Velocity ……Page 77
2.6 Short-Range Case ……Page 79
2.7 Long-Range Case ……Page 82
2.8 The Eikonal Equation ……Page 93
2.9 Smoothness of Trajectories ……Page 94
3.0 Introduction ……Page 98
3.1 Time-Dependent Schrodinger Hamiltonians ……Page 102
3.2 Asymptotic Momentum ……Page 103
3.3 Fast-Decaying Case ……Page 109
3.4 Slow-Decaying Case – Hormander Potentials ……Page 111
3.5 Slow-Decaying Case – Smooth Potentials ……Page 120
3.6 Dollard Wave Operators ……Page 123
3.7 Isozaki-Kitada Construction ……Page 125
3.8.1 Adiabatic evolution ……Page 129
3.8.2 Counterexample Based on the Adiabatic Approximation ……Page 130
3.8.3 A Sharper Counterexample ……Page 132
3.9 Smoothness of Wave Operators in the Fast-Decaying Case ……Page 134
3.10 Smoothness of Wave Operators in the Slow-Decaying Case ……Page 137
4.0 Introduction ……Page 140
4.1 Schrodinger Hamiltonians ……Page 148
4.2 Weak Large Velocity Estimates ……Page 150
4.3 The Mourre Estimate and its Consequences ……Page 153
4.4 Asymptotic Velocity ……Page 156
4.5 Joint Spectrum of $P^+$ and $H$ ……Page 166
4.6 Short-Range Case ……Page 169
4.7 Long-Range Case ……Page 172
4.8 Dollard Wave Operators ……Page 179
4.9 The Isozaki-Kitada Construction ……Page 181
4.10.1 The Born-Oppenheimer Approximation – an Abstract Setting ……Page 186
4.10.2 The Born-Oppenheimer Approximation for Schrodinger Operators ……Page 188
4.10.3 Counterexample to Asymptotic Completeness ……Page 191
4.11 Strong Large Velocity Estimates ……Page 195
4.12 Strong Propagation Estimates for the Generator of Dilations ……Page 198
4.13 Strong Low Velocity Estimates ……Page 201
4.14 Schrodinger Operators as Pseudo-differential Operators ……Page 203
4.15 Improved Isozaki-Kitada Modifiers ……Page 204
4.16 Microlocal Propagation Estimates ……Page 208
4.17 Wave Operators with Outgoing Cutoffs ……Page 212
4.18 Wave Operators on Weighted Spaces ……Page 214
5.0 Introduction ……Page 220
5.1 $N$-Body Systems ……Page 224
5.2 Some Special Observables ……Page 231
5.3 Bounded Trajectories and the Classical Mourre Estimate ……Page 242
5.4 Asymptotic Velocity ……Page 248
5.5 Joint Localization of the Energy and the Asymptotic Velocity ……Page 253
5.6 Regular $a$—Trajectories ……Page 254
5.7 Upper Bound on the Size of Clusters ……Page 258
5.8.1 Short-Range Free Region Case ……Page 261
5.8.2 Long-Range Free Region Case ……Page 262
5.9.2 Asymptotic External Position in the Long-range Case ……Page 264
5.10 Potentials of Super-Exponential Decay ……Page 267
6.0 Introduction ……Page 270
6.1 Basic Definitions ……Page 279
6.2 HVZ Theorem ……Page 281
6.3 Weak Large Velocity Estimates ……Page 285
6.4 The Mourre Estimate ……Page 286
6.5 Exponential Decay of Eigenfunctions and Absence of Positive Eigenvalues ……Page 294
6.6 Asymptotic Velocity ……Page 302
6.7 Asymptotic Completeness of Short-Range Systems ……Page 311
6.8 Asymptotic Separation of the Dynamics I ……Page 314
6.9 Time-Dependent $N$-Body Hamiltonians ……Page 320
6.10 Joint Spectrum of $P^+$ and $H$ ……Page 324
6.11 Asymptotic Clustering and Asymptotic Absolute Continuity ……Page 331
6.12 Improved Propagation Estimates ……Page 333
6.13 Upper Bound on the Size of Clusters ……Page 338
6.14 Asymptotic Separation of the Dynamics II ……Page 350
6.15 Modified Wave Operators and Asymptotic Completeness in the Long-Range Case ……Page 352
A.1 Some Inequalities ……Page 358
A.2 The Fixed Point Theorem ……Page 361
A.3 The Hamilton-Jacobi Equation ……Page 365
A.4 Construction of Some Cutoff Functions ……Page 372
A.5 Propagation Estimates ……Page 373
A.6 Comparison of Two Dynamics ……Page 374
A.7 Schwartz’s Global Inversion Theorem ……Page 377
B.1 Self-Adjoint Operators ……Page 378
B.2 Convergence of Self-Adjoint Operators ……Page 381
B.3 Time-Dependent Hamiltonians ……Page 384
B.4 Propagation Estimates ……Page 388
B.5 Limits of Unitary Operators ……Page 390
B.6 Schur’s Lemma ……Page 391
B.7 Compact Operators in $L^2(mathbb{R}^n)$ ……Page 392
C.1 Basic Estimates of Commutators ……Page 394
C.2 Almost-Analytic Extensions ……Page 395
C.3 Commutator Expansions I ……Page 397
C.4 Commutator Expansions II ……Page 399
D.0 Introduction ……Page 402
D.1 Symbols of Operators ……Page 404
D.2 Phase Space Correlation Functions ……Page 405
D.3 Symbols Associated with a Uniform Metric ……Page 406
D.4 Pseudo-differential Operators Associated with a Uniform Metric ……Page 408
D.5 Symbols and Operators Depending on a Parameter ……Page 412
D.7 Symbols Associated with some Non-Uniform Metrics ……Page 415
D.8 Pseudo-differential Operators Associated with the Metric $g_1$ ……Page 416
D.9 Essential Support of Pseudo-differential Operators ……Page 419
D.10 Ellipticity ……Page 421
D.11 Functional Calculus for Pseudo-differential Operators Associated with the Metric $g_1$ ……Page 423
D.12 Non-Stationary Phase Method ……Page 426
D.13 FIO’s Associated with a Uniform Metric ……Page 427
D.15 FIO’s Associated with the Metric $g_1$ ……Page 430
References ……Page 438
Index ……Page 448
Table of Contents ……Page 450