Demuth / Witt / Schulze Partial Differential Equations and Spectral Theory

1. Auflage 2011
ISBN: 978-3-0348-0024-2
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 341 Seiten

Reihe: Advances in Partial Differential Equations

This volume collects six articles on selected topics at the frontier between partial differential equations and spectral theory, written by leading specialists in their respective field. The articles focus on topics that are in the center of attention of current research, with original contributions from the authors. They are written in a clear expository style that makes them accessible to a broader audience. The articles contain a detailed introduction and discuss recent progress, provide additional motivation, and develop the necessary tools. Moreover, the authors share their views on future developments, hypotheses, and unsolved problems.

Autoren/Hrsg.

Witt, Ingo

Schulze, Bert-Wolfgang

Demuth, Michael

Inhaltsverzeichnis

1;Contents;6
2;Preface;10
3;Quantum Semiconductor Models;12
3.1;1. Introduction;12
3.1.1;1.1. A first example;12
3.1.2;1.2. Structure of the paper;14
3.2;2. Derivation of the models;15
3.2.2;2.2. Quantum drift diffusion equations;18
3.2.3;2.3. Viscous quantum hydrodynamics;20
3.2.4;2.4. Historical background and further models;23
3.3;3. The quantum drift diffusion model;25
3.3.1;3.1. Introduction;25
3.3.1.2;3.1.2. Questions and problems.;27
3.3.1.3;3.1.3. Methods.;28
3.3.2;3.2. A special fourth-order parabolic equation;29
3.3.2.1;3.2.1. The one-dimensional case.;30
3.3.2.2;3.2.2. The two- and three-dimensional cases.;32
3.3.3;3.3. Quantum drift diffusion equations in one dimension;33
3.3.3.1;3.3.1. Global weak solution.;33
3.3.3.2;3.3.2. Semiclassical limit e . 0.;35
3.3.3.3;3.3.3. Quasineutral limit . . 0.;38
3.3.3.4;3.3.4. Long time behavior.;39
3.3.4;3.4. Quantum drift diffusion equations in two and three dimensions;40
3.3.5;3.5. Entropy based methods;40
3.3.5.1;3.5.1. Approximate problems.;41
3.3.5.2;3.5.2. Entropy inequalities.;42
3.3.5.3;3.5.3. Compactness argument.;44
3.3.5.3.1;Global weak solution;44
3.3.5.3.2;Semiclassical limit;45
3.3.5.3.3;Quasineutral Limit;46
3.3.5.4;3.5.4. Long time asymptotics.;47
3.3.6;3.6. Open problems;48
3.4;4. The viscous quantum hydrodynamic model;48
3.4.1;4.1. Known results;48
3.4.2;4.2. Main results;51
3.4.3;4.3. Elliptic systems of mixed order;53
3.4.3.1;4.3.1. General results.;53
3.4.3.2;4.3.2. Mixed-order systems in quantum hydrodynamics.;60
3.4.4;4.4. Stationary states and their stability;68
3.4.4.1;4.4.1. Geometric results.;69
3.4.4.2;4.4.2. Application to the viscous quantum hydrodynamic system. ;71
3.4.4.2.1;Appendix: A variant of Aubin’s lemma;74
3.5;Acknowledgment;76
3.6;References;76
4;Large Coupling Convergence: Overview and New Results;84
4.1;1. Introduction;84
4.2;2. Non-negative form perturbations;86
4.2.1;2.1. Notation and general hypotheses;86
4.2.2;2.2. A resolvent formula;89
4.2.3;2.3. Convergence with respect to the operator norm;90
4.2.4;2.4. Schr¨odinger operators;96
4.2.5;2.5. Convergence within a Schatten-von Neumann class;100
4.2.6;2.6. Compact perturbations;103
4.2.6.1;2.6.1. Expansions.;103
4.2.6.2;2.6.2. Schatten-von Neumann classes.;106
4.2.7;2.7. Dynkin’s formula;108
4.2.8;2.8. Differences of powers of resolvents;111
4.3;3. Dirichlet forms;115
4.3.1;3.1. Notation and basic results;115
4.3.2;3.2. Trace of a Dirichlet form;117
4.3.3;3.3. A domination principle;122
4.3.4;3.4. Convergence with maximal rate and equilibrium measures;123
4.4;Acknowledgment;127
4.5;References;127
5;Smooth Spectral Calculus;129
5.1;1. Introduction;130
5.2;2. Functional spaces and notation;133
5.3;3. The basic abstract structure;134
5.3.1;3.1. The limiting absorption prinicple – LAP;136
5.3.2;3.2. Persistence of smoothness under functional operations;140
5.4;4. Short-range perturbations;141
5.4.1;4.1. The exceptional set SP;143
5.5;5. Sums of tensor products;149
5.5.1;5.1. The operator H = H1 I2 + I1 H2;150
5.5.2;5.2. Extending the abstract framework of the LAP;152
5.5.3;5.3. The LAP for H = H1 I2 + I1 H2;153
5.5.4;5.4. The Stark Hamiltonian;156
5.5.5;5.5. The operator H0 = -. and some wild perturbations;159
5.6;6. The limiting absorption principle for second-order divergence-type operators;162
5.6.1;6.1. The operator H0 = -. – revisited;164
5.6.2;6.2. Proof of the LAP for the operator H;167
5.6.3;6.3. An application: Existence and completeness of the wave operators W±(H,H0);173
5.7;7. An eigenfunction expansion theorem;173
5.8;8. Global spacetime estimates for a generalized wave equation;180
5.9;9. Further directions and open problems;186
5.9.1;1. Estimating the heat kernel in Lebesgue spaces;186
5.9.2;2. Abstract approach to long-range perturbations;187
5.9.3;3. Discreteness of eigenvalues of short-range perturbations;187
5.9.4;4. High energy estimates of divergence-type operators;188
5.10;References;188
6;Spectral Analysis and Geometry of Sub-Laplacian and Related Grushin-type Operators;193
6.1;1. Introduction;194
6.2;2. Sub-Riemannian manifolds;198
6.3;3. Bicharacteristic flow of Grushin-type operator;202
6.4;4. Heisenberg group case;206
6.4.1;4.1. Grushin-type operators;206
6.4.2;4.2. Isoperimetric interpretation and double fibration: Grushin plane case;208
6.5;5. Sub-Riemannian structure on SL(2, R);212
6.5.1;5.1. A sub-Riemannian structure and Grushin-type operator;212
6.5.2;5.2. Horizontal curves: SL(2, R);216
6.5.3;5.3. Isoperimetric interpretation: SL(2, R) . Upper half-plane;218
6.6;6. The S3 . P1(C) case;219
6.6.1;6.1. Spherical Grushin operator and Grushin sphere;219
6.6.2;6.2. Geodesics on the Grushin sphere;222
6.7;7. Quaternionic structure on R8 and sub-Riemannian structures;230
6.7.1;7.1. Vector fields on S7 and sub-Riemannian structures;231
6.7.2;7.2. Hopf fibration and a sub-Riemannian structure;234
6.7.3;7.3. Singular metric on S4 and a spherical Grushin operator;236
6.7.4;7.4. Sub-Riemannian structure on a hypersurface in S7;238
6.8;8. Sub-Riemannian structure on nilpotent Lie groups;241
6.9;9. Engel group and Grushin-type operators;242
6.9.1;9.1. Engel group and their subgroups;242
6.9.2;9.2. Solution of a Hamilton-Jacobi equation;245
6.10;10. Free two-step nilpotent Lie algebra and group;249
6.11;11. 2-step nilpotent Lie groups of dimension = 6;250
6.11.1;11.1. Heat kernel of the free nilpotent Lie group of dimension 6;250
6.11.2;11.2. Heat kernel of Grushin-type operators;253
6.12;12. Spectrum of a five-dimensional compact nilmanifold;260
6.13;13. Spectrum of a six-dimensional compact nilmanifold;273
6.14;14. Heat trace asymptotics on compact nilmanifolds of the dimensions five and six;274
6.14.1;14.1. The six-dimensional case;274
6.14.2;14.2. The five-dimensional case;279
6.15;15. Concluding remarks;281
6.15.1;Appendix A. Basic theorems for pseudo-differential operators of Weyl symbols and heat kernel construction;282
6.15.2;Appendix B. Heat kernel of the sub-Laplacian on 2-step nilpotent groups;287
6.15.3;Appendix C. The trace of the fundamental solution;291
6.15.4;Appendix D. Selberg trace formula;295
6.16;Acknowledgment;297
6.17;References;297
7;Zeta Functions of Elliptic Cone Operators;301
7.1;1. Introduction;301
7.2;2. Classical results;302
7.3;3. Conical singularities;304
7.4;4. Cone differential operators;307
7.5;5. Domains;311
7.6;6. Spectra;316
7.7;7. Rays of minimal growth for elliptic cone operators;318
7.8;8. Asymptotics;323
7.9;References;328
8;Pseudodifferential Operators on Manifolds: A Coordinate-free Approach;331
8.1;1. Introduction;331
8.2;2. PDOs: local definition and basic properties;332
8.3;3. Linear connections;334
8.4;4. PDOs: a coordinate-free approach;337
8.5;5. Functions of the Laplacian;340
8.6;6. An approximate spectral projection;343
8.7;7. Other known results and possible developments;345
8.7.2;7.2. Operators on sections of vector bundles;346
8.7.3;7.3. Noncompact manifolds;347
8.7.4;7.4. Other symbol classes;347
8.7.5;7.5. Operators generated by vector fields;347
8.7.6;7.6. Operators on Lie groups;348
8.7.7;7.7. Geometric aspects and physical applications;348
8.8;References;349