Spectral theory: basic concepts and applications
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham, Switzerland
Springer Nature Switzerland AG
[2020]
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| Schriftenreihe: | Graduate texts in mathematics
284 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | x, 338 Seiten Illustrationen |
| ISBN: | 9783030380014 |
Internformat
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Datensatz im Suchindex
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| adam_text | Contents 1 Introduction......................................................................................................... 1 2 Hilbert Spaces..................................................................................................... 5 5 7 9 15 18 24 27 31 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Normed Vector Spaces........................................................................... Lp Spaces................................................................................................ Bounded Linear Maps............................................................................ Hilbert Spaces......................................................................................... Sobolev Spaces....................................................................................... Orthogonality.......................................................................................... Orthonormal Bases................................................................................. Exercises.................................................................................................. 3 Operators............................................................................................................. 3.1 3.2 3.3 3.4 3.5 3.6 Unbounded Operators............................................................................ Adjoints................................................................................................... Closed Operators.................................................................................... Symmetry and Self-
adjointness............................................................. Compact Operators................................................................................. Exercises.................................................................................................. 4 Spectrum and Resolvent .................................................................................. 4.1 4.2 4.3 4.4 4.5 5 Definitions and Examples...................................................................... Resolvent ................................................................................................ Spectrum of Self-adjoint Operators....................................................... Spectral Theory of Compact Operators ............................................... Exercises.................................................................................................. 35 35 37 41 47 57 62 67 67 79 86 89 96 101 Unitary Operators ................................................................................... 102 The Main Theorem................................................................................. 107 Functional Calculus............................................................................... 112 Spectral Decomposition......................................................................... 115 Exercises.................................................................................................. 121 The Spectral Theorem....................................................................................... 5.1 5.2 5.3 5.4 5.5 ix
Contents x 6 The Laplacian with Boundary Conditions.................................................. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 125 Self-adjoint Extensions........................................................................... 129 Discreteness of Spectrum...................................................................... 135 Regularity of Eigenfunctions................................................................ 143 Eigenvalue Computations...................................................................... 147 Asymptotics of Dirichlet Eigenvalues.................................................. 155 Nodal Domains....................................................................................... 171 Isoperimetric Inequalities and Minimal Eigenvalues.......................... 174 Exercises.................................................................................................. 179 7 Schrôdinger Operators..................................................................................... 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Positive Potentials.................................................................................. Relatively Bounded Perturbations......................................................... Relatively Compact Perturbations......................................................... Hydrogen Atom ..................................................................................... Semiclassical Asymptotics..................................................................... Periodic
Potentials.................................................................................. Exercises.................................................................................................. 183 184 194 197 203 207 214 220 8 Operators on Graphs......................................................................................... 225 Combinatorial Laplacians...................................................................... Quantum Graphs...................................................................................... Spectral Properties of Compact Quantum Graphs............................... Eigenvalue Comparison......................................................................... Eigenvalue Asymptotics......................................................................... Exercises.................................................................................................. 226 230 232 234 237 242 9 Spectral Theory on Manifolds......................................................................... 245 245 250 262 266 270 282 287 291 298 8.1 8.2 8.3 8.4 8.5 8.6 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Smooth Manifolds................................................................................... Riemannian Metrics............................................................................... The Laplacian.......................................................................................... Spectrum of a Compact Manifold......................................................... Heat
Equation.......................................................................................... Wave Propagation on Compact Manifolds........................................... Complete Manifolds and Essential Self-adjointness........................... Essential Spectrum of Complete Manifolds......................................... Exercises.................................................................................................. 303 Measure and Integration ........................................................................ 303 Lp Spaces................................................................................................ 315 Fourier Transform.................................................................................. 320 Elliptic Regularity.................................................................................. 324 A Background Material........................................................................................ A. 1 A.2 A.3 A.4 References................................................................................................................... 331 Index............................................................................................................................. 335
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| adam_txt |
Contents 1 Introduction. 1 2 Hilbert Spaces. 5 5 7 9 15 18 24 27 31 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Normed Vector Spaces. Lp Spaces. Bounded Linear Maps. Hilbert Spaces. Sobolev Spaces. Orthogonality. Orthonormal Bases. Exercises. 3 Operators. 3.1 3.2 3.3 3.4 3.5 3.6 Unbounded Operators. Adjoints. Closed Operators. Symmetry and Self-
adjointness. Compact Operators. Exercises. 4 Spectrum and Resolvent . 4.1 4.2 4.3 4.4 4.5 5 Definitions and Examples. Resolvent . Spectrum of Self-adjoint Operators. Spectral Theory of Compact Operators . Exercises. 35 35 37 41 47 57 62 67 67 79 86 89 96 101 Unitary Operators . 102 The Main Theorem. 107 Functional Calculus. 112 Spectral Decomposition. 115 Exercises. 121 The Spectral Theorem. 5.1 5.2 5.3 5.4 5.5 ix
Contents x 6 The Laplacian with Boundary Conditions. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 125 Self-adjoint Extensions. 129 Discreteness of Spectrum. 135 Regularity of Eigenfunctions. 143 Eigenvalue Computations. 147 Asymptotics of Dirichlet Eigenvalues. 155 Nodal Domains. 171 Isoperimetric Inequalities and Minimal Eigenvalues. 174 Exercises. 179 7 Schrôdinger Operators. 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Positive Potentials. Relatively Bounded Perturbations. Relatively Compact Perturbations. Hydrogen Atom . Semiclassical Asymptotics. Periodic
Potentials. Exercises. 183 184 194 197 203 207 214 220 8 Operators on Graphs. 225 Combinatorial Laplacians. Quantum Graphs. Spectral Properties of Compact Quantum Graphs. Eigenvalue Comparison. Eigenvalue Asymptotics. Exercises. 226 230 232 234 237 242 9 Spectral Theory on Manifolds. 245 245 250 262 266 270 282 287 291 298 8.1 8.2 8.3 8.4 8.5 8.6 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Smooth Manifolds. Riemannian Metrics. The Laplacian. Spectrum of a Compact Manifold. Heat
Equation. Wave Propagation on Compact Manifolds. Complete Manifolds and Essential Self-adjointness. Essential Spectrum of Complete Manifolds. Exercises. 303 Measure and Integration . 303 Lp Spaces. 315 Fourier Transform. 320 Elliptic Regularity. 324 A Background Material. A. 1 A.2 A.3 A.4 References. 331 Index. 335 |
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| author | Borthwick, David |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
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| institution | BVB |
| isbn | 9783030380014 |
| language | English |
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| spelling | Borthwick, David Verfasser (DE-588)1112155821 aut Spectral theory basic concepts and applications David Borthwick Cham, Switzerland Springer Nature Switzerland AG [2020] © 2020 x, 338 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 284 Partial Differential Equations Operator Theory Functional Analysis Partial differential equations Operator theory Functional analysis Spektraltheorie (DE-588)4116561-5 gnd rswk-swf Unbeschränkter Operator (DE-588)4236037-7 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Differentialoperator (DE-588)4012251-7 gnd rswk-swf Spektraltheorie (DE-588)4116561-5 s Unbeschränkter Operator (DE-588)4236037-7 s Hilbert-Raum (DE-588)4159850-7 s Differentialoperator (DE-588)4012251-7 s DE-604 Erscheint auch als Online-Ausgabe 978-3-030-38002-1 Graduate texts in mathematics 284 (DE-604)BV000000067 284 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032709317&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borthwick, David Spectral theory basic concepts and applications Graduate texts in mathematics Partial Differential Equations Operator Theory Functional Analysis Partial differential equations Operator theory Functional analysis Spektraltheorie (DE-588)4116561-5 gnd Unbeschränkter Operator (DE-588)4236037-7 gnd Hilbert-Raum (DE-588)4159850-7 gnd Differentialoperator (DE-588)4012251-7 gnd |
| subject_GND | (DE-588)4116561-5 (DE-588)4236037-7 (DE-588)4159850-7 (DE-588)4012251-7 |
| title | Spectral theory basic concepts and applications |
| title_auth | Spectral theory basic concepts and applications |
| title_exact_search | Spectral theory basic concepts and applications |
| title_exact_search_txtP | Spectral theory basic concepts and applications |
| title_full | Spectral theory basic concepts and applications David Borthwick |
| title_fullStr | Spectral theory basic concepts and applications David Borthwick |
| title_full_unstemmed | Spectral theory basic concepts and applications David Borthwick |
| title_short | Spectral theory |
| title_sort | spectral theory basic concepts and applications |
| title_sub | basic concepts and applications |
| topic | Partial Differential Equations Operator Theory Functional Analysis Partial differential equations Operator theory Functional analysis Spektraltheorie (DE-588)4116561-5 gnd Unbeschränkter Operator (DE-588)4236037-7 gnd Hilbert-Raum (DE-588)4159850-7 gnd Differentialoperator (DE-588)4012251-7 gnd |
| topic_facet | Partial Differential Equations Operator Theory Functional Analysis Partial differential equations Operator theory Functional analysis Spektraltheorie Unbeschränkter Operator Hilbert-Raum Differentialoperator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032709317&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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