Analysis as a Tool in Mathematical Physics - In Memory of Boris Pavlov

In Memoriam......Page 6
Contents......Page 9
Part I Boris Pavlov: Life and Mathematics......Page 13
Positions Held......Page 15
Research Interests......Page 16
List of PhD Students......Page 17
Publications by Boris Pavlov......Page 18
Photographs (private and academic life)......Page 28
Academician A.N. Terenin’s rules for researchers......Page 36
Water-colours by Boris Pavlov......Page 37
Pavlov’s Mathematics......Page 41
My highest achievements are......Page 42
Zero-range potentials with internal structure and solvable models......Page 43
References......Page 46
Pavlov’s perturbations......Page 47
References......Page 49
Boris Pavlov and bases, as I remember them......Page 51
1. First step – uniform proximity......Page 52
2. Second step – generating and sine-type functions......Page 53
3. Third step – B. Pavlov’s idea......Page 54
4. Post-Pavlov developments, unconditional bases......Page 55
5. Generalizations and applications......Page 57
References......Page 59
Prelude......Page 62
Some background......Page 63
A pair of contractive analytic functions that are unimodular on the boundary; the vector representation of Hardy and Lebesgue spaces......Page 64
The main results......Page 65
Some related developments......Page 66
References......Page 67
Part II Research Papers......Page 71
1. Introduction......Page 72
2. Reminder of resolvents’ basic properties......Page 74
3. M.G. Krein’s line of argument......Page 77
4. Singular perturbations of selfadjoint Laplace operator. Null-range potentials......Page 79
5. Singular perturbations of selfadjoint Laplace operator. 1D-located perturbation......Page 83
References......Page 87
1. Introduction......Page 89
2. Resonances as zeroes of a characteristic determinant......Page 92
3. Absence of cancellations in Leibniz formula......Page 94
References......Page 100
1. Introduction......Page 103
2.1. Sturm–Liouville problem......Page 107
2.2. Asymmetric spaces associated to wave equation......Page 110
3.1. An associated moment problem......Page 112
3.2. The special case......Page 114
4.1. Divided differences......Page 117
5.1. The reachable set......Page 119
References......Page 126
1. Statement of the problem......Page 129
2. Strategy for the evaluation of the second Weyl coefficient......Page 132
3. The Weyl symbol of the resolvent......Page 135
4. The matrix trace of the resolvent......Page 137
5. The matrix trace of a power of the resolvent......Page 138
6. Asymptotic expansion for the density f......Page 139
7. The second Weyl coefficient......Page 140
Appendix A. Proof of Lemma 2.2......Page 143
Appendix B. Some integrals involving the functions gn......Page 147
Appendix C. Proof of Lemma 2.3......Page 149
Appendix D. Weyl quantization on manifolds......Page 150
Appendix E. The resolvent and its powers as pseudodifferential operators......Page 153
Appendix F. Proof of formulae (4.4) and (4.5)......Page 159
References......Page 160
1. Introduction......Page 163
2. Main Result......Page 165
References......Page 169
1. Introduction......Page 171
2. Scattering systems and Weyl functions of quasi boundary triples......Page 172
3. Main result......Page 176
4. Proof of Theorem 3.1......Page 178
5. An example......Page 184
References......Page 189
1.1. Preliminaries......Page 192
1.4. The quantum Rabi Hamiltonian......Page 193
2.2. Comments......Page 194
3.1. Reduction to Jacobi matrices......Page 195
3.2. Bogoliubov transformation and GRWA......Page 196
3.3. Approximation of et(↖â)......Page 197
3.5. Sketch of the proof of (GRWA)......Page 198
4. How the parameters can be recovered from the spectrum......Page 199
References......Page 201
1. Introduction......Page 203
2. Extension theory and boundary triples......Page 206
3. Formulae for the resolvents of almost solvable extensions......Page 209
4. Functional model and theorems about smooth vectors......Page 212
5. The relationship between the set of smooth vectors and the absolutely continuous subspace in the self-adjoint setting......Page 216
6. Wave and scattering operators......Page 220
7. Spectral representation for the absolutely continuous part of the operator A0......Page 225
Acknowledgements......Page 227
Appendix......Page 228
References......Page 235
1. Introduction......Page 240
2. The Real Case......Page 242
3. The Complex Case......Page 244
4. Equality in a Totik–Widom Upper Bound......Page 247
5. On a Theorem of Erdős......Page 251
6. Invariance of Widom Factors Under Polynomial Preimages......Page 252
Acknowledgments......Page 253
References......Page 254
1. Introduction......Page 256
2. Statement of results......Page 258
3. The normal form......Page 266
4. The eikonal equation......Page 276
5. The transport equations......Page 277
6. The proof of Theorem 2.11......Page 281
References......Page 284
1. Introduction......Page 286
2. Notation......Page 290
3. The Weyl quantization......Page 291
4. Product #......Page 292
5. Quantum non-degenerate matrices......Page 293
6. Oscillator semigroup......Page 295
7. Positive elements of the oscillator semigroup......Page 297
9. Relationship between and symplectic group......Page 299
11. Polar decomposition......Page 302
12. Trace and the trace norm......Page 304
13. Operator norm......Page 305
14. One degree of freedom......Page 306
15. Application to the boundedness of pseudo-differential operators......Page 307
16. Complex symplectic Lie algebra......Page 309
17. Hamiltonians......Page 310
18. Holomorphic 1-parameter subsemigroups......Page 311
References......Page 312
1. Exterior and interior dynamics of tectonic plates......Page 314
2. Modelling the resonance interaction between SGO and beating phenomena......Page 321
3. Example: A circular active zone......Page 323
4. Resonance conditions for circular plates......Page 327
5. A simple model of alternation......Page 330
6. Appendix 1: Natural boundary conditions and the perturbed biharmonic wave equation......Page 337
References......Page 341
1. Introduction......Page 344
2. Some Background Material......Page 345
3. On Positivity Preserving Linear Operators in Lp(Rn)m......Page 349
References......Page 358
1. Introduction and Main Results......Page 360
1.1. Counting paths in graphs......Page 361
1.2. Weighted random walks on graphs......Page 362
2.2. Perron’s projection......Page 364
2.3. Comparison between the non-weighted case and the weighted case......Page 366
3.2. The Laplace Transform of the counting and probability functions......Page 367
4. Proof of main results......Page 369
5.1. Summation over regions of Pascal triangle......Page 375
5.2. Multiscale Substitution Schemes......Page 376
5.3. Physics Applications......Page 378
References......Page 379
1. Introduction......Page 382
2. Saddle point forms......Page 385
3. Reducing subspaces......Page 390
4. Regular embeddings and direct rotations......Page 394
5. Block-diagonalization of associated operators by a direct rotation......Page 396
6. The Riccati equation......Page 398
7. Some applications......Page 402
Acknowledgements......Page 406
References......Page 407
1. Introduction......Page 410
2. Linear Operator......Page 413
3. Proof of The Main Result......Page 417
References......Page 424
1. The Akhiezer Integral Transforms: a formal definition......Page 426
2. The operators Cω and Sω in L2......Page 430
3. The Akhiezer operators Φω and Ψω in L2 ⊕ L2......Page 434
4. The Hilbert transform......Page 436
5. The operators Cω and Sω in L2σ......Page 438
6. The Akhiezer operators Φω and Ψω in L2σ ⊕ L2σ......Page 441
References......Page 442
1. Introduction......Page 443
2.1. Finite range operators and perturbation determinants......Page 444
2.2. Geronimo{Case equations......Page 446
2.3. Locations of resonances and eigenvalues......Page 448
3. Random matrices coupled to the Laplacian......Page 449
References......Page 455
1. Introduction and physical background......Page 457
2. Acceptable functional parameters φ(x) and the general spectral structure of the corresponding operators  Kφ......Page 462
3.1. Dimension 1: Fourier transform, the adjoint operator, and the selfadjointness in the weighted space......Page 465
3.2. Dimension 1: Reference operator......Page 466
3.3. Dimension 1: Spectral structure and spectral estimations for the operator Kφ corresponding to an acceptable equilibrium distribution function φ(x)......Page 469
3.4. Dimension 1: the Gaussian equilibrium distributions......Page 473
3.5. Dimension 1: Quadratic form approach, links to the mean-value functionals and a new property of the Legendre polynomials......Page 476
4.1. Dimensions M - 2: General scheme, completely admissible domains and reduction to a countable set of 1D problems......Page 478
4.2. Spectral estimations for the operators Kφ and  Qφ, corresponding to an acceptable pseudo-toroidal symmetric functions to an acceptable pseudo-toroidal symmetric functions φ(r) having completely admissible support Ω = bTN ⸦ RM......Page 481
4.3. Planar case and application to the thin film relaxation processes......Page 485
4.4. Application to the matter relaxation processes in cylindric 3D domains......Page 488
4.5. Application to the matter relaxation processes in and toroidal 3D domains......Page 489
5. New class of special functions generated by the operators Kφ in higher dimensions......Page 490
5.2. Rank-1 special functions......Page 491
5.3. Rank-2 special functions......Page 495
6.1. Infinite Jacobi matrices with unbounded entries......Page 497
7. Inverse problems (reconstruction of the external field φ(x) through the spectrum of the operator Kφ) in 1D case......Page 498
8. Conclusions and further tasks......Page 503
References......Page 505
1. Introduction and background......Page 510
Definition 1.......Page 511
Definition 2.......Page 512
3. The inverse problem for integral-difference operators in simple graphs......Page 513
Theorem 1.......Page 515
References......Page 516
1. Introduction......Page 520
2. Geometry of the filter: preliminaries......Page 522
3. Boundary conditions......Page 525
4. Scattering matrix via Intermediate Hamiltonian......Page 529
5. Re-normalization of spectral data of the Schrödinger operator on the quantum well......Page 533
6. Appendix: the intermediate Hamiltonian......Page 543
References......Page 546
1. Introduction......Page 549
2.1. Quantum graph model......Page 550
2.2. Scattering, functional model and completeness criterion......Page 552
3. Proof of the main theorem......Page 554
3.1. Relation between magnetic and non-magnetic cases......Page 555
3.2. Graph containing balanced vertices......Page 557
3.3. Graph containing only unbalanced vertices......Page 558
References......Page 559
Appendix......Page 561
1. Introduction......Page 563
2. Content......Page 565
3. Notation and preliminaries......Page 566
4.1. Projective quasi-norms and approximation properties......Page 567
4.2. Spectral type......Page 570
4.3.-extension property and-lifting property......Page 572
4.4. Applications. Zd-symmetry for N[r,p] and N[r,p]......Page 574
4.6. Sharpness of main results......Page 575
References......Page 577
1. Introduction......Page 579
2. Preliminaries......Page 580
3. An ordering result for the lowest mixed eigenvalues of polygons......Page 581
References......Page 583
1. Introduction......Page 585
2. Boundary Value Problems by Example......Page 590
3. Spectral Boundary Value Problem and its M-function......Page 602
4. Boundary Conditions......Page 608
5. Linear Operators of Boundary Value Problems......Page 612
6. Cayley Transform of M-function. Applications to the scattering theory......Page 621
7. Singular Perturbations......Page 625
Acknowledgements......Page 629
References......Page 630