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Partial Differential Equations and Spectral Theory 2011 ed. [Hardback]

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  • Formāts: Hardback, 341 pages, weight: 758 g, X, 341 p., 1 Hardback
  • Sērija : Operator Theory: Advances and Applications 211
  • Izdošanas datums: 02-Feb-2011
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3034800231
  • ISBN-13: 9783034800235
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Partial Differential Equations and Spectral Theory 2011 ed.Palielināt
  • Formāts: Hardback, 341 pages, weight: 758 g, X, 341 p., 1 Hardback
  • Sērija : Operator Theory: Advances and Applications 211
  • Izdošanas datums: 02-Feb-2011
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3034800231
  • ISBN-13: 9783034800235
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783034800235.html
Keywords:
In a clear, expository style, this book offers seven articles on topics at the frontier of partial differential equations and spectral theory. The authors discuss recent progress and share their views on future developments, hypotheses and unsolved problems.

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.
Preface ix
Quantum Semiconductor Models
1(72)
L. Chen
M. Dreher
1 Introduction
1(3)
1.1 A first example
1(2)
1.2 Structure of the paper
3(1)
2 Derivation of the models
4(10)
2.1 Quantum Vlasov and quantum Boltzmann equations
4(3)
2.2 Quantum drift diffusion equations
7(2)
2.3 Viscous quantum hydrodynamics
9(3)
2.4 Historical background and further models
12(2)
3 The quantum drift diffusion model
14(23)
3.1 Introduction
14(4)
3.2 A special fourth order parabolic equation
18(4)
3.3 Quantum drift diffusion equations in one dimension
22(7)
3.4 Quantum drift diffusion equations in two and three dimensions
29(1)
3.5 Entropy based methods
29(8)
3.6 Open problems
37(1)
4 The viscous quantum hydrodynamic model
37(26)
4.1 Known results
37(3)
4.2 Main results
40(2)
4.3 Elliptic systems of mixed order
42(15)
4.4 Stationary states and their stability
57(6)
Appendix: A variant of Aubin's lemma
63(2)
Acknowledgment
65(1)
References
65(8)
Large Coupling Convergence: Overview and New Results
73(46)
H. BelHadjAli
A. Ben Amor
J.F. Brasche
1 Introduction
73(2)
2 Non-negative form perturbations
75(29)
2.1 Notation and general hypotheses
75(3)
2.2 A resolvent formula
78(1)
2.3 Convergence with respect to the operator norm
79(6)
2.4 Schrodinger operators
85(4)
2.5 Convergence within a Schatten-von Neumann class
89(3)
2.6 Compact perturbations
92(5)
2.7 Dynkin's formula
97(3)
2.8 Differences of powers of resolvents
100(4)
3 Dirichlet forms
104(12)
3.1 Notation and basic results
104(2)
3.2 Trace of a Dirichlet form
106(5)
3.3 A domination principle
111(1)
3.4 Convergence with maximal rate and equilibrium measures
112(4)
Acknowledgment
116(1)
References
116(3)
Smooth Spectral Calculus
119(64)
M. Ben-Artzi
1 Introduction
120(3)
2 Functional spaces and notation
123(1)
3 The basic abstract structure
124(7)
3.1 The limiting absorption prinicple -- LAP
126(4)
3.2 Persistence of smoothness under functional operations
130(1)
4 Short-range perturbations
131(8)
4.1 The exceptional set ΣP
133(6)
5 Sums of tensor products
139(13)
5.1 The operator H = H1 ⊗ I2 + I1 ⊗ H2
140(2)
5.2 Extending the abstract framework of the LAP
142(1)
5.3 The LAP for H = H1 ⊗ I2 + I1 ⊗ H2
143(3)
5.4 The Stark Hamiltonian
146(3)
5.5 The operator H0 = -Δ and some wild perturbations
149(3)
6 The limiting absorption principle for second-order divergence-type operators
152(11)
6.1 The operator H0 = -Δ - revisited
154(3)
6.2 Proof of the LAP for the operator H
157(6)
6.3 An application: Existence and completeness of the wave operators W ± (H, H0)
163(1)
7 An eigenfunction expansion theorem
163(7)
8 Global spacetime estimates for a generalized wave equation
170(6)
9 Further directions and open problems
176(2)
References
178(5)
Spectral Analysis and Geometry of Sub-Laplacian and Related Grushin-type Operators
183(108)
W. Bauer
K. Furutani
C. Iwasaki
1 Introduction
184(4)
2 Sub-Riemannian manifolds
188(4)
3 Bicharacteristic flow of Grushin-type operator
192(4)
4 Heisenberg group case
196(6)
4.1 Grushin-type operators
196(2)
4.2 Isoperimetric interpretation and double fibration: Grushin plane case
198(4)
5 Sub-Riemannian structure on SL(2, R)
202(7)
5.1 A sub-Riemannian structure and Grushin-type operator
202(4)
5.2 Horizontal curves: SL(2, R)
206(2)
5.3 Isoperimetric interpretation: SL(2, R) → Upper half-plane
208(1)
6 The S3 → P1(C) case
209(11)
6.1 Spherical Grushin operator and Grushin sphere
209(3)
6.2 Geodesics on the Grushin sphere
212(8)
7 Quaternionic structure on R8 and sub-Riemannian structures
220(11)
7.1 Vector fields on S7 and sub-Riemannian structures
221(3)
7.2 Hopf fibration and a sub-Riemannian structure
224(2)
7.3 Singular metric on S4 and a spherical Grushin operator
226(2)
7.4 Sub-Riemannian structure on a hypersurface in S7
228(3)
8 Sub-Riemannian structure on nilpotent Lie groups
231(1)
9 Engel group and Grushin-type operators
232(7)
9.1 Engel group and their subgroups
232(3)
9.2 Solution of a Hamilton-Jacobi equation
235(4)
10 Free two-step nilpotent Lie algebra and group
239(1)
11 2-step nilpotent Lie groups of dimension ≤ 6
240(10)
11.1 Heat kernel of the free nilpotent Lie group of dimension 6
240(3)
11.2 Heat kernel of Grushin-type operators
243(7)
12 Spectrum of a five-dimensional compact nilmanifold
250(13)
13 Spectrum of a six-dimensional compact nilmanifold
263(1)
14 Heat trace asymptotics on compact nilmanifolds of the dimensions five and six
264(7)
14.1 The six-dimensional case
264(5)
14.2 The five-dimensional case
269(8)
15 Concluding remarks
271(260)
Appendix A Basic theorems for pseudo-differential operators of Wey1 symbols and heat kernel construction
272(5)
Appendix B Heat kernel of the sub-Laplacian on 2-step nilpotent groups
277(4)
Appendix C The trace of the fundamental solution
281(4)
Appendix D Selberg trace formula
285(2)
Acknowledgment
287(1)
References
287(4)
Zeta Functions of Elliptic Cone Operators
291(30)
G.A. Mendoza
1 Introduction
291(1)
2 Classical results
292(2)
3 Conical singularities
294(3)
4 Cone differential operators
297(4)
5 Domains
301(5)
6 Spectra
306(2)
7 Rays of minimal growth for elliptic cone operators
308(5)
8 Asymptotics
313(5)
References
318(3)
Pseudodifferential Operators on Manifolds: A Coordinate-free Approach
321
P. McKeag
Y. Safarov
1 Introduction
321(1)
2 PDOs: local definition and basic properties
322(2)
3 Linear connections
324(3)
4 PDOs: a coordinate-free approach
327(3)
5 Functions of the Laplacian
330(3)
6 An approximate spectral projection
333(2)
7 Other known results and possible developments
335(4)
7.1 Other definitions for scalar PDOs
335(1)
7.2 Operators on sections of vector bundles
336(1)
7.3 Noncompact manifolds
337(1)
7.4 Other symbol classes
337(1)
7.5 Operators generated by vector fields
337(1)
7.6 Operators on Lie groups
338(1)
7.7 Geometric aspects and physical applications
338(1)
7.8 Global phase functions
338(1)
References
339