Partial Differential Equations and Spectral Theory (eBook)

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