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Spectral Theory - Basic Concepts and Applications 2020 book

Spectral Theory - Basic Concepts and Applications

Details Of The Book

Spectral Theory - Basic Concepts and Applications

Category: performance analysis
edition: 1 
Authors:   
serie: Graduate Texts in Mathematics 
ISBN : 9783030380014, 9783030380021 
publisher: Springer 
publish year: 2020 
pages: 339 
language: English 
ebook format : PDF (It will be converted to PDF, EPUB OR AZW3 if requested by the user) 
file size: 4 MB 

price : $12.92 17 With 24% OFF



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Abstract Of The Book



Table Of Contents

Preface......Page 7
Contents......Page 9
1 Introduction......Page 11
Notes......Page 13
2.1 Normed Vector Spaces......Page 14
2.2 Lp Spaces......Page 16
2.3 Bounded Linear Maps......Page 18
2.3.1 Operator Topologies......Page 20
2.3.2 Uniform Boundedness......Page 22
2.4 Hilbert Spaces......Page 24
2.5.1 Weak Derivatives......Page 27
2.5.2 Hm Spaces......Page 30
2.6 Orthogonality......Page 33
2.7 Orthonormal Bases......Page 36
2.7.1 Weak Sequential Compactness......Page 39
2.8 Exercises......Page 40
Notes......Page 42
3.1 Unbounded Operators......Page 43
3.2 Adjoints......Page 45
3.2.1 Adjoints of Unbounded Operators......Page 46
3.3 Closed Operators......Page 49
3.3.1 Closable Operators......Page 50
3.3.2 Closed Graph Theorem......Page 52
3.3.3 Invertibility......Page 54
3.4 Symmetry and Self-adjointness......Page 55
3.4.1 Self-adjoint Operators......Page 56
3.4.2 Criteria for Self-adjointness......Page 60
3.4.3 Friedrichs Extension......Page 63
3.5 Compact Operators......Page 65
3.5.1 Hilbert–Schmidt Operators......Page 68
3.6 Exercises......Page 70
Notes......Page 72
4.1 Definitions and Examples......Page 74
4.1.1 Basic Properties of the Spectrum......Page 76
4.1.2 Spectrum of a Multiplication Operator......Page 77
4.1.3 Resolvent of the Euclidean Laplacian......Page 79
4.1.4 Discrete Laplacians......Page 81
4.2.1 Analytic Operator-Valued Functions......Page 86
4.2.2 Analyticity of the Resolvent......Page 90
4.2.3 Spectral Radius......Page 91
4.3 Spectrum of Self-adjoint Operators......Page 93
4.4 Spectral Theory of Compact Operators......Page 96
4.4.1 Spectral Theorem for Compact Self-adjoint Operators......Page 99
4.4.2 Hilbert–Schmidt Operators......Page 100
4.4.3 Traces......Page 102
4.5 Exercises......Page 103
Notes......Page 105
5 The Spectral Theorem......Page 107
5.1 Unitary Operators......Page 108
5.1.1 Continuous Functional Calculus......Page 109
5.1.2 Spectral Measures......Page 111
5.1.3 Spectral Theorem for Unitary Operators......Page 112
5.2 The Main Theorem......Page 113
5.3 Functional Calculus......Page 118
5.4 Spectral Decomposition......Page 121
5.4.1 Discrete and Essential Spectrum......Page 122
5.4.2 Continuous Spectrum......Page 124
5.4.3 The Min–Max Principle......Page 125
5.5 Exercises......Page 127
Notes......Page 128
6 The Laplacian with Boundary Conditions......Page 130
6.1.1 The Space H10(Ω)......Page 134
6.1.2 The Dirichlet Laplacian......Page 136
6.1.3 The Neumann Laplacian......Page 139
6.2 Discreteness of Spectrum......Page 140
6.2.1 Periodic Sobolev Spaces......Page 142
6.2.2 Extension Lemmas......Page 143
6.3 Regularity of Eigenfunctions......Page 148
6.4 Eigenvalue Computations......Page 152
6.4.1 Finite Element Method......Page 154
6.4.2 Domain Monotonicity......Page 155
6.4.3 Neumann Eigenvalues......Page 158
6.5 Asymptotics of Dirichlet Eigenvalues......Page 160
6.5.1 Strategy for the Proof......Page 162
6.5.2 Asymptotics of the Resolvent Kernel......Page 163
6.5.3 Trace Asymptotics......Page 168
6.5.4 The Tauberian Argument......Page 172
6.6 Nodal Domains......Page 176
6.7 Isoperimetric Inequalities and Minimal Eigenvalues......Page 179
6.8 Exercises......Page 184
Notes......Page 187
7 Schrödinger Operators......Page 188
7.1 Positive Potentials......Page 189
7.1.1 Essential Self-adjointness......Page 190
7.1.2 Quadratic Form Extension......Page 191
7.1.3 Discrete Spectrum......Page 193
7.1.4 Quantum Harmonic Oscillator......Page 195
7.2 Relatively Bounded Perturbations......Page 199
7.3 Relatively Compact Perturbations......Page 202
7.4 Hydrogen Atom......Page 208
7.5 Semiclassical Asymptotics......Page 212
7.6.1 Floquet Theory......Page 219
7.6.2 Spectrum of H......Page 222
7.7 Exercises......Page 225
Notes......Page 228
8 Operators on Graphs......Page 229
8.1 Combinatorial Laplacians......Page 230
8.2 Quantum Graphs......Page 234
8.3 Spectral Properties of Compact Quantum Graphs......Page 236
8.4 Eigenvalue Comparison......Page 238
8.5 Eigenvalue Asymptotics......Page 241
8.5.1 Weyl Law......Page 245
8.6 Exercises......Page 246
Notes......Page 247
9.1 Smooth Manifolds......Page 248
9.1.1 Tangent and Cotangent Vectors......Page 250
9.1.2 Partition of Unity......Page 252
9.2 Riemannian Metrics......Page 253
9.2.1 Geodesics and the Exponential Map......Page 256
9.2.2 Completeness......Page 264
9.3 The Laplacian......Page 265
9.3.1 Green\'s Identity......Page 267
9.4 Spectrum of a Compact Manifold......Page 269
9.4.1 Dirichlet Eigenvalues......Page 271
9.4.2 Regularity......Page 272
9.5 Heat Equation......Page 273
9.5.1 Maximum Principle......Page 276
9.5.2 Heat Kernel......Page 277
9.5.3 Spectral Applications......Page 283
9.6 Wave Propagation on Compact Manifolds......Page 285
9.6.1 Propagation Speed......Page 288
9.7 Complete Manifolds and Essential Self-adjointness......Page 290
9.8 Essential Spectrum of Complete Manifolds......Page 294
9.8.1 Decomposition Principle......Page 295
9.8.2 The Bottom of the Essential Spectrum......Page 296
9.8.3 Volume Growth Estimate......Page 298
9.9 Exercises......Page 301
Notes......Page 303
A.1 Measure and Integration......Page 305
A.1.1 Lebesgue Measure......Page 306
A.1.2 Integration......Page 307
A.1.4 Differentiation......Page 309
A.1.5 Decomposition of Measures......Page 311
A.1.6 Riesz Representation......Page 312
A.2 Lp Spaces......Page 317
A.2.1 Completeness......Page 319
A.2.2 Convolution......Page 321
A.3 Fourier Transform......Page 322
A.4 Elliptic Regularity......Page 326
References......Page 332
Index......Page 336


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