Bolotnikov / Rodman Topics in Operator Theory

Volume 2: Systems and Mathematical Physics
1. Auflage 2011
ISBN: 978-3-0346-0161-0
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 446 Seiten

This is the second volume of a collection of original and review articles on recent advances and new directions in a multifaceted and interconnected area of mathematics and its applications. It encompasses many topics in theoretical developments in operator theory and its diverse applications in applied mathematics, physics, engineering, and other disciplines. The purpose is to bring in one volume many important original results of cutting edge research as well as authoritative review of recent achievements, challenges, and future directions in the area of operator theory and its applications.

Autoren/Hrsg.

Bolotnikov, Vladimir

Rodman, Leiba

Inhaltsverzeichnis

1;Title Page;4
2;Copyright Page;5
3;Table of Contents;6
4;The XIXth International Workshop on Operator Theory and its Applications. II;8
5;Exact Solutions to the Nonlinear Schrodinger Equation;11
5.1;1. Introduction;11
5.2;2. Main results;14
5.3;3. Examples;18
5.4;References;21
6;Robust Control, Multidimensional Systems and Multivariable Nevanlinna-Pick Interpolation;23
6.1;1. Introduction;23
6.2;2. The 1-D systems/single-variable case;28
6.2.1;2.1. The model-matching problem;30
6.2.2;2.2. The frequency-domain stabilization and H8 problem;30
6.2.3;2.3. The state-space approach;32
6.2.4;2.4. Notes;35
6.3;3. The fractional representation approach to stabilizability and performance;35
6.3.1;3.1. Parametrization of stabilizing controllers in terms of a given stabilizing controller;37
6.3.2;3.2. The Youla-Kucera parametrization;41
6.3.3;3.3. The standard H8-problem reduced to model matching.;45
6.3.4;3.4. Notes;47
6.4;4. Feedback control for linear time-invariant multidimensional systems;48
6.4.1;4.1. Multivariable frequency-domain formulation;48
6.4.2;4.2. Multidimensional state-space formulation;51
6.4.3;4.3. Equivalence of frequency-domain and state-space formulations;62
6.5;5. Robust control with structured uncertainty: the commutative case;68
6.5.1;5.1. Gain-scheduling in state-space coordinates;69
6.5.2;5.2. Gain-scheduling: a pure frequency-domain formulation;71
6.5.3;5.3. Robust control with a hybrid frequency-domain/state-space formulation;71
6.5.4;5.4. Notes;73
6.6;6. Robust control with dynamic time-varying structured uncertainty;74
6.6.1;6.1. The state-space LFT-model formulation;74
6.6.2;6.2. A noncommutative frequency-domain formulation;79
6.6.3;6.3. Equivalence of state-space noncommutative LFT-model and noncommutative frequency-domain formulation;84
6.6.4;6.4. Notes;88
6.6.5;References;90
7;Absence of Existence and Uniqueness for Forward-backward Parabolic Equations on a Half-line;99
7.1;1. Introduction;99
7.2;2. Preliminaries;101
7.3;3. Absence of existence and uniqueness;102
7.4;4. Satisfaction of (C1)–(C4);105
7.5;References;107
8;Bounds for Eigenvalues of the p-Laplacian with Weight Function of Bounded Variation;109
8.1;1. Introduction;109
8.2;2. Generalized total variation;111
8.3;3. Estimates of eigenvalues;113
8.4;4. Optimality of bounds I;116
8.5;5. Optimality of bounds II;117
8.6;6. The periodic p-Laplacian;120
8.7;References;122
9;The Gelfand-Levitan Theory for Strings;124
9.1;1. Introduction;124
9.2;2. Notation;127
9.3;3. The spectral function .0,ß;128
9.4;4. The transformation operator;131
9.5;5. Existence of the transformation operator;134
9.6;6. Smoothness;139
9.7;7. The inverse problem;139
9.8;8. The limit-point case;141
9.9;9. The string;142
9.10;References;144
10;On the Uniqueness of a Solution to Anisotropic Maxwell’s Equations;146
10.1;Introduction;146
10.2;1. Basic boundary value problems for Maxwell’s equations;147
10.3;2. A fundamental solution to Maxwell’s operator;153
10.4;3. Green’s formulae;160
10.5;4. Representation of solutions and layer potentials;162
10.6;5. The uniqueness of a solution;165
10.7;References;172
11;Dichotomy and Boundedness of Solutions for Some Discrete Cauchy Problems;174
11.1;1. Introduction;174
11.2;2. Preliminary results;175
11.3;3. Dichotomy and boundedness;177
11.4;4. The case of operators acting on Banach spaces;180
11.5;References;182
12;Control Laws for Discrete Linear Repetitive Processes with Smoothed Previous Pass Dynamics;184
12.1;1. Introduction;184
12.2;2. Preliminaries and the new model;186
12.3;3. Stability analysis;188
12.4;4. Stabilization;191
12.5;5. Robustness;193
12.6;6. Numerical example;198
12.7;7. Conclusions and further work;200
12.8;References;201
13;Fourier Method for One-dimensional Schrodinger Operators with Singular Periodic Potentials;203
13.1;1. Introduction;203
13.2;2. Preliminary results;206
13.3;3. Fourier representation of the operators LPer±;221
13.4;4. Fourier representation for the Hill–Schrodinger operator with Dirichlet boundary conditions;224
13.5;5. Localization of spectra;230
13.6;6. Conclusion;239
13.7;References;241
14;Additive Invariants on Quantum Channels and Regularized Minimum Entropy;245
14.1;1. Introduction;245
14.2;2. Proof of Theorem 1.1;247
14.3;3. Examples;248
14.4;4. Bi-quantum channels;251
14.5;References;252
15;A Functional Model, Eigenvalues, and Finite Singular Critical Points for Indefinite Sturm-Liouville Operators;254
15.1;1. Introduction;254
15.2;2. The functional model for indefinite Sturm-Liouville operators with one turning point;257
15.2.1;2.1. Preliminaries; the functional model of a symmetric operator257
15.2.2;2.2. The functional model for J-self-adjoint extensions of symmetric operators;258
15.2.3;2.3. The Sturm-Liouville case;263
15.3;3. Point and essential spectra of the model operator A and of indefinite Sturm-Liouville operators;267
15.3.1;3.1. Point spectrum of the model operator;267
15.3.2;3.2. Essential and discrete spectra of the model operator and of indefinite Sturm-Liouville operators;277
15.3.3;3.3. Non-emptiness of resolvent set for Sturm-Liouville operators;279
15.4;4. The absence of embedded eigenvalues and other applications;280
15.4.1;4.1. The absence of embedded eigenvalues for the case of infinite-zone potentials;280
15.4.2;4.2. Other applications;283
15.5;5. Remarks on indefinite Sturm-Liouville operators with the singular critical point 0;284
15.6;6. Discussion;286
15.7;Appendix A. Boundary triplets for symmetric operators;289
15.8;References;290
16;On the Eigenvalues of the Lax Operator for the Matrix-valued AKNS System;295
16.1;1. Introduction;295
16.2;2. The AKNS differential operator;298
16.3;3. Operators with symmetries;299
16.4;4. Square-integrable solutions on a half-line;303
16.5;5. Jost solutions;308
16.6;6. Bounds on the location of eigenvalues;314
16.7;7. Nonexistence of eigenvalues;315
16.8;8. Nonexistence of purely imaginary eigenvalues;319
16.9;9. Purely imaginary eigenvalues;320
16.10;References;327
17;An Extension Theorem for Bounded Forms Defined in Relaxed Discrete Algebraic Scattering Systems and the Relaxed Commutant Lifting Theorem;330
17.1;1. Introduction;330
17.2;2. Preliminaries;332
17.3;3. Relaxed discrete algebraic scattering systems;335
17.4;4. Interpolants of relaxed discrete algebraic scattering systems and forms defined on them;336
17.5;5. An extension theorem for bounded forms defined in relaxed discrete algebraic scattering systems;337
17.5.1;5.1. The extension theorem for bounded forms defined in relaxed discrete alge- braic scattering systems generalizes the Cotlar-Sadosky Extension Theorem;343
17.5.2;5.2. The extension theorem for bounded forms defined in relaxed discrete alge- braic scattering systems generalizes the Relaxed Commuting Lifting Theorem;345
17.6;References;348
18;Deconstructing Dirac Operators. III: Dirac and Semi-Dirac Pairs;351
18.1;1. Introduction;351
18.2;2. Pairs of first-order differential operators;354
18.2.1;2.1. Prerequisites;354
18.2.2;2.2. Spherical means;355
18.2.3;2.3. Related integral operators;356
18.2.4;2.4. Integral representation formulas;357
18.3;3. Auxiliary results and proofs;358
18.3.1;3.1. An integral formula;358
18.3.2;3.2. Two lemmas;359
18.3.3;3.3. Proofs of Theorems A and B;360
18.4;4. Concluding remarks;361
18.4.1;4.1. Refining Theorem A;361
18.4.2;4.2. Refining Theorem B;362
18.5;References;364
19;Mapping Properties of Layer Potentials Associated with Higher-order Elliptic Operators in Lipschitz Domains;367
19.1;1. Introduction;367
19.2;2. Preliminaries;372
19.2.1;2.1. Lipschitz domains and nontangential maximal function;372
19.2.2;2.2. Smoothness spaces in Rn;373
19.2.3;2.3. Smoothness spaces in Lipschitz domains;374
19.2.4;2.4. Smoothness spaces on Lipschitz boundaries;376
19.3;3. Function spaces of Whitney arrays;380
19.3.1;3.1. Whitney-Lebesgue, Whitney-Sobolev and Whitney-Besov spaces;380
19.3.2;3.2. Whitney-Hardy and Whitney-Triebel-Lizorkin spaces;381
19.3.3;3.3. Multi-trace theory;382
19.3.4;3.4. Whitney-BMO and Whitney-VMO spaces;386
19.4;4. The double layer potential;387
19.4.1;4.1. The fundamental solution;387
19.4.2;4.2. Definition and nontangential maximal estimates;389
19.4.3;4.3. Jump relations;395
19.4.4;4.4. Estimates on Besov and Triebel-Lizorkin spaces;398
19.5;5. The single layer operator;400
19.5.1;5.1. Definition and nontangential maximal estimates;400
19.5.2;5.2. Estimates on Besov and Triebel-Lizorkin spaces;403
19.5.3;5.3. The conormal derivative;403
19.5.4;5.4. Jump relations for the conormal derivative;407
19.6;References;409
20;Applications of a Numerical Spectral Expansion Method to Problems in Physics; a Retrospective412
20.1;1. Introduction;412
20.2;2. The spectral expansion method (IEM);413
20.3;3. Numerical properties of the method;418
20.4;4. Retrospective of the work with Israel Koltracht;422
20.5;5. Subsequent developments;423
20.6;6. Summary and conclusions;424
20.7;Appendix A;425
20.8;References;428
21;Regularized Perturbation Determinants and KdV Conservation Laws for Irregular Initial Profiles;430
21.1;1. Introduction;430
21.2;2. Notation and preliminaries;432
21.3;3. Regularized perturbation determinants;433
21.4;4. The regularized perturbation 2-determinant is a KdV invariant;437
21.5;5. Almost conserved quantities;441
21.6;6. Applications to spectral theory of the Schrodinger operator;442
21.7;7. Appendix: Impedance form of Schr¨odinger operators with singular potentials;445
21.8;References;446