Frontiers in Analysis and Probability In the Spirit of the Strasbourg-Zürich Meetings

Preface......Page 5
Contents......Page 6
1 Introduction......Page 7
2.1 Random Planar Waves......Page 10
2.2 Points of Isotropic Scaling......Page 12
3 Number of Critical Points and Size of the Zero Set......Page 14
3.1 Global Statistics......Page 15
3.2.1 Local Universality of Zeros......Page 17
3.2.2 Local Universality of Critical Points......Page 19
4.1 Diffeomorphism Types......Page 20
4.2 Nesting Configurations......Page 22
References......Page 25
1 Introduction—Comments on the Foundations of Quantum Mechanics and Purpose of Paper......Page 27
2 Standard Formulation of Quantum Mechanics and Its Shortcomings......Page 29
3 Summary of the ``ETH-Approach''......Page 33
4 Scattered Remarks About Indirect Measurements, Conclusions......Page 47
References......Page 50
1 Introduction......Page 52
1.1 Conventions and Preliminaries......Page 53
1.2 p-Cohomology in Degree One......Page 54
1.3 Some Examples......Page 55
2.1 Quasi-Isometries......Page 58
2.2 Boundaries......Page 59
2.2.2 Boundary of Hyperbolic Spaces......Page 60
2.2.3 ``Nonlinear'' Boundaries......Page 61
2.3 Representation Theory......Page 62
3 1-Cohomology and the Ends......Page 63
4 p-Cohomology and Harmonic Functions......Page 66
4.1 Reduction to Bounded Functions......Page 68
4.2 Values at Infinity......Page 69
4.3 Harmonic Functions......Page 72
5.1 Further Results......Page 74
5.2 Questions......Page 76
References......Page 79
1 Introduction......Page 82
2 The Dirichlet Problem and Global λ-Polyharmonic Functions......Page 85
3 Boundary Value Problems for λ-Polyharmonic Functions......Page 87
4 Comparison with the Case of Infinite Trees; Examples......Page 90
References......Page 95
1 Introduction: The Model and the Main Results......Page 96
1.1 Interacting Electrons......Page 97
1.2 The Ground State Energy Per Particle......Page 99
1.3 The Ground State: Its One- and Two-Particle Density Matrices......Page 101
1.3.2 The Non-interacting Ground State......Page 102
1.3.3 The Interacting Ground State......Page 104
1.4 Discussion and Perspectives......Page 107
1.4.1 Why the Pieces Model?......Page 108
1.4.2 Outline of the Paper......Page 109
2.1 Rescaling the Operator......Page 111
2.1.1 Other Asymptotic Regimes......Page 112
2.2 The Analysis of the One-Particle Pieces Model......Page 113
2.3 The Statistics of the Pieces......Page 114
2.4 Free Electrons......Page 115
2.4.1 The Ground State Energy Per Particle......Page 116
3.1 Decomposition by Occupation Numbers......Page 117
3.2 Almost Sure Non-degeneracy of the Interacting Ground State......Page 120
3.3 The Proof of Theorem 1.1......Page 121
3.4 The Approximate Ground State opt......Page 122
3.5 Comparing opt with the Ground State of the Interacting System......Page 127
3.6 The Analysis of HUp......Page 128
3.7 The Proof of Theorem 1.3......Page 152
4.1 From the Occupation Decomposition to the Reduced Density Matrices......Page 154
4.1.1 The One-Particle Density Matrix......Page 155
4.1.2 The Two-Particle Density Matrix......Page 156
4.1.3 A Particular Case......Page 161
4.2 The Proof of Theorem 1.5......Page 165
4.3 The Proof of Theorem 1.6......Page 179
5 Almost Sure Convergence for the Ground State Energy Per Particle......Page 191
6.1.1 The Proof of Proposition 1.4......Page 198
6.1.2 The Proof of Lemma 6.1......Page 202
6.1.3 The Ground State of Two Interacting Electrons and Its Density Matrices......Page 213
6.2 Electrons in Distinct Pieces......Page 215
6.3 The Proof of Lemma 4.11......Page 221
A.1 Facts on the Poisson Process......Page 222
A.2 The Proof of Proposition 2.2......Page 223
A.3 The Proof of Propositions 2.3 and 2.4......Page 225
A.4 The Proof of Proposition 2.5......Page 228
A.5 The Proof of Proposition 2.6......Page 229
Appendix B A Simple Lemma on Trace Class Operators......Page 230
Appendix C Anti-symmetric Tensors: The Projector on Anti-symmetric Functions......Page 231
D.1 Proof of the One-Particle Density Matrix Reduction, Theorem 4.2......Page 233
D.2 Proof of the Two-Particle Density Matrix Reduction, Theorem 4.4......Page 236
References......Page 246
Entropies for Negatively Curved Manifolds......Page 248
1 Local Symmetry and Volume Growth......Page 249
2.1 Dynamical Entropy......Page 253
2.2 Thermodynamical Formalism......Page 254
2.3 Liouville Measure......Page 256
3.1 Patterson–Sullivan......Page 257
3.2 Bowen–Margulis......Page 259
3.3 Burger–Roblin......Page 260
4.1 Foliated Diffusions......Page 263
4.2 Stable Diffusions......Page 265
4.3  Stochastic Entropy and Rigidity......Page 268
5.1 Stochastic Flow Adapted to Lρ......Page 272
5.3 Continuity of the Relative Entropy......Page 274
5.5 Conclusion. Katok's Conjecture......Page 276
References......Page 277
Introduction......Page 280
1.1 The Holonomy Process and the Yang–Mills Action......Page 281
1.2.1 The Yang–Mills Action......Page 283
1.2.2 Gauge Transformations......Page 285
1.2.3 Some Questions of Classical Yang–Mills Theory......Page 286
1.2.4 The Holonomy Map......Page 287
1.3 The Yang–Mills Action: Holonomies......Page 289
1.3.1 Holonomies......Page 290
1.3.3 The Yang–Mills Action......Page 291
1.4.1 The Configuration Space of Lattice Yang–Mills Theory......Page 292
1.4.2 The Driver–Sengupta Formula......Page 293
1.4.3 Invariance Under Subdivision......Page 295
1.4.4 The Continuum Limit......Page 296
1.4.5 The Structure of the Holonomy Process......Page 298
1.5 Wilson Loop Expectations......Page 301
2.1 The Brownian Motion on the Unitary Group......Page 303
2.2.1 Using Harmonic Analysis......Page 305
2.2.2 Using Stochastic Calculus......Page 307
2.3 Yin . . .......Page 308
2.4 . . . And Yang......Page 310
2.5 The Case of the Sphere: A Not So Simple Loop......Page 313
3.1 First Approach......Page 318
3.2 Makeenko and Migdal's Proof......Page 319
3.3 The Equations, Their Merits and Demerits......Page 323
3.4 The Master Field on Compact Surfaces......Page 325
3.5 A Value of the Master Field on the Plane......Page 327
References......Page 328
Notation......Page 331
1 Introduction......Page 332
2 Preliminary Results from MNNAOP......Page 336
3 A Flow of Operators on the Space Generated by the Limiting Eigenvectors......Page 340
4 An Inner Product on the Domain of the Operators......Page 352
5 A More Intrinsic Definition of the Limiting Flow of Operators......Page 360
References......Page 372
1 Derivation of Gibbs Measures for NLS from Many-Body Quantum Systems......Page 374
1.1 The Stationary Problem......Page 375
1.2 The Time-Dependent Problem......Page 381
References......Page 384
1 Introduction......Page 386
References......Page 389
2  The Basic Linear Models......Page 390
2.1 Schrödinger Representation of the Isotropic Harmonic Oscillator......Page 391
2.1.1 Wigner Distributions......Page 393
2.1.2 Weyl Pseudo-Differential Operators, Metaplectic Covariance......Page 394
3  Bargmann-Fock Space and the Toeplitz Representation of the Isotropic Oscillator......Page 395
3.1 Lifting to the Heisenberg Group......Page 397
3.2  Metaplectic Representation......Page 399
3.3 Toeplitz Construction of the Metaplectic Representation......Page 400
3.4 Toeplitz Quantization of Hamiltonian Flows......Page 401
3.5 Bargmann Intertwining Operator Between Schrödinger and Bargmann-Fock......Page 402
3.6  Analogies and Correspondences Between the Real and Complex Settings......Page 403
4.1 Allowed and Forbidden Regions and the Caustic......Page 405
4.2 Scaling Asymptotics Around the Caustic in Physical Space......Page 406
4.3 Nodal Sets of Random Hermite Eigenfunctions......Page 407
4.5  The Kac–Rice Formula......Page 410
5 Interfaces in Phase Space for Schrödinger Operators: Wigner Distributions......Page 411
5.1 Interface Asymptotics for Wigner Distributions of Individual Eigenspace Projections......Page 412
5.2  Interior Bessel Asymptotics......Page 413
5.4 Exterior Asymptotics......Page 414
5.5  Supremum at ρ=0......Page 415
5.6 Sums of Eigenspace Projections......Page 416
5.7  Interior Asymptotics for .12em.1emdotteddotteddotted.76dotted.6h-Localized Weyl Sums......Page 417
5.8  Interface Asymptotics for Smooth .12em.1emdotteddotteddotted.76dotted.6h2/3-Localized Weyl Sums......Page 418
5.9 Sharp .12em.1emdotteddotteddotted.76dotted.6h2/3-Localized Weyl Sums......Page 419
5.10  Bulk Sums......Page 420
5.12 Interface Asymptotics for Bulk Weyl Sums of Definition 5.13......Page 421
5.13  Heuristics......Page 422
5.14 Remark on Nodal Sets in Phase Space......Page 423
6  Interfaces in Phase Space: Partial Bergman Kernels......Page 424
6.2 Unrescaled Bulk Results on dμkz......Page 429
6.3 k-Scaling Results on d μz, 1/2k......Page 430
6.4 Energy Level Localization and d μz, 1,αk......Page 431
6.5 Critical Levels......Page 432
6.6 Interface Asymptotics at Critical Levels......Page 433
6.7 Sketch of Proof......Page 435
7  Interfaces for the Bargmann-Fock Isotropic Harmonic Oscillator......Page 436
8  Bargmann-Fock Space of a Line Bundle and Interface Asymptotics......Page 437
8.1 Volume Forms......Page 438
8.2 Orthonormal Basis......Page 439
8.3 Bargmann-Fock Bergman Kernel of a Line Bundle......Page 440
8.4 Interface Asymptotics......Page 441
9.1  Further Types of Interface Problems......Page 443
9.2 Droplets in Phase Space......Page 444
10.1 Holomorphic Sections in Lk and CR-Holomorphic Functions on X......Page 445
10.3 Boutet de Monvel-Sjöstrand Parametrix for the Szegö Kernel......Page 447
10.4 Lifting the Hamiltonian Flow to a Contact Flow on Xh......Page 448
11.2 Appendix on Laguerre Functions......Page 449
References......Page 450