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

Description

Boris Pavlov (1936-2016), to whom this volume is dedicated, was a prominent specialist in analysis, operator theory, and mathematical physics. As one of the most influential members of the St. Petersburg Mathematical School, he was one of the founders of the Leningrad School of Non-self-adjoint Operators. This volume collects research papers originating from two conferences that were organized in memory of Boris Pavlov: “Spectral Theory and Applications”, held in Stockholm, Sweden, in March 2016, and “Operator Theory, Analysis and Mathematical Physics – OTAMP2016” held at the Euler Institute in St. Petersburg, Russia, in August 2016. The volume also includes water-color paintings by Boris Pavlov, some personal photographs, as well as tributes from friends and colleagues. 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