Spectral theory of infinite-volume hyperbolic surfaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2007
|
| Schriftenreihe: | PM - Progress in Mathematics
282 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XI, 355 S. Ill., graph. Darst. |
| ISBN: | 9780817645243 0817645241 |
Internformat
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| 100 | 1 | |a Borthwick, David |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Spectral theory of infinite-volume hyperbolic surfaces |c David Borthwick |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XI, 355 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a PM - Progress in Mathematics |v 282 | |
| 650 | 4 | |a Riemann, Surfaces de | |
| 650 | 4 | |a Spectre (Mathématiques) | |
| 650 | 4 | |a Riemann surfaces | |
| 650 | 4 | |a Spectral theory (Mathematics) | |
| 650 | 0 | 7 | |a Hyperbolische Fläche |0 (DE-588)4735194-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Spektraltheorie |0 (DE-588)4116561-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hyperbolische Fläche |0 (DE-588)4735194-9 |D s |
| 689 | 0 | 1 | |a Spektraltheorie |0 (DE-588)4116561-5 |D s |
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Datensatz im Suchindex
| _version_ | 1805088672857456640 |
|---|---|
| adam_text |
Contents
1
Introduction
. 1
2
Hyperbolic Surfaces
. 7
2.1
The hyperbolic plane
. 8
2.2
Fuchsian groups
. 13
2.3
Geometrically finite groups
. 18
2.4
Classification of hyperbolic ends
. 22
2.5
Gauss-Bonnet theorem
. 28
2.6
Length spectrum and Selberg's
zeta
function
. 31
3
Compact and Finite-Area Surfaces
. 37
3.1
Selberg's trace formula for compact surfaces
. 37
3.2
Consequences of the trace formula
. 42
3.3
Finite-area hyperbolic surfaces
. 45
4
Spectral Theory for the Hyperbolic Plane
. 49
4.1
Resolvent
. 49
4.2
Generalized eigenfimctions
. 52
4.3
Scattering matrix
. 56
5
Model Resolvents for Cylinders
. 61
5.1
Hyperbolic cylinders
. 61
5.2
Funnels
. 68
5.3
Parabolic cylinder
. 70
6
The Resolvent
. 75
6.1
Compactification
. 75
6.2
Analytic
Fredholm
theorem
. 79
6.3
Continuation of the resolvent
. 81
6.4
Structure of the resolvent kernel
. 84
6.5
The stretched product
. 87
x
Contents
7
Spectral
and Scattering Theory
. 93
7.1
Essential and discrete spectrum
. 93
7.2
Absence of embedded eigenvalues
. 95
7.3
Generalized eigenfunctions
. 102
7.4
Scattering matrix
. 105
7.5
Scattering matrices for the runnel and cylinders
. 114
8
Resonances and Scattering Poles
. 117
8.1
Multiplicities of resonances
. 118
8.2
Structure of the resolvent at a resonance
. 119
8.3
Scattering poles
. 124
8.4
Operator logarithmic residues
. 126
8.5
Half-integer points
. 131
8.6
Coincidence of resonances and scattering poles
. 137
9
Upper Bound for Resonances
. 147
9.1
Resonances and zeros of determinants
. 148
9.2
Singular value estimates
. 151
9.3
Upper bound
. 154
9.4
Estimates on model terms
. 156
10
Selberg
Zeta
Function
. 171
10.1
Relative scattering determinant
. 173
10.2
Regularized traces
. 175
10.3
The resolvent 0-trace calculation
. 183
10.4
Structure of the
zeta
function
. 189
10.5
Order bound
. 196
10.6
Determinant of the Laplacian
. 203
11
Wave Trace and
Poisson
Formula
. 207
1
1.1
Regularized wave trace
. 208
11.2
Model wave kernel
. 209
11.3
Wave 0-trace formula
. 211
11.4
Poisson
formula
. 215
12
Resonance Asymptotics
. 223
12.1
Lower bound on resonances
. 223
12.2
Lower bound near the critical line
. 226
12.3
Weyl formula for the scattering phase
. 229
13
Inverse Spectral Geometry
. 237
13.1
Resonances and the length spectrum
. 238
13.2
Hyperbolic trigonometry
. 239
13.3 Teichmüller
space
. 242
13.4
Finiteness of
isospectral
classes
. 248
Contents xi
14
Patterson-Sullivan Theory
. 259
14.1
A measure on the limit set
. 259
14.2
Ergodicity
. 267
14.3
Hausdorff measure of the limit set
. 274
14.4
The first resonance
. 278
14.5
Prime geodesic theorem
. 284
14.6
Refined asymptotics of the length spectrum
.289
15
Dynamical Approach to the
Zeta
Function
. 297
15.1
Schottky groups
. 298
15.2
Symbolic dynamics
. 300
15.3
Dynamical
zeta
function
. 303
15.4
Growth estimates
. 308
A Appendix
. 315
A.I Entire functions
. 315
A.2 Distributions and Fourier transforms
. 320
A.3 Spectral theory
. 324
A.4 Singular values, traces, and determinants
. 330
A.5 Pseudodifferential operators
. 336
References
. 341
Notation Guide
. 351
Index
. 353 |
| adam_txt |
Contents
1
Introduction
. 1
2
Hyperbolic Surfaces
. 7
2.1
The hyperbolic plane
. 8
2.2
Fuchsian groups
. 13
2.3
Geometrically finite groups
. 18
2.4
Classification of hyperbolic ends
. 22
2.5
Gauss-Bonnet theorem
. 28
2.6
Length spectrum and Selberg's
zeta
function
. 31
3
Compact and Finite-Area Surfaces
. 37
3.1
Selberg's trace formula for compact surfaces
. 37
3.2
Consequences of the trace formula
. 42
3.3
Finite-area hyperbolic surfaces
. 45
4
Spectral Theory for the Hyperbolic Plane
. 49
4.1
Resolvent
. 49
4.2
Generalized eigenfimctions
. 52
4.3
Scattering matrix
. 56
5
Model Resolvents for Cylinders
. 61
5.1
Hyperbolic cylinders
. 61
5.2
Funnels
. 68
5.3
Parabolic cylinder
. 70
6
The Resolvent
. 75
6.1
Compactification
. 75
6.2
Analytic
Fredholm
theorem
. 79
6.3
Continuation of the resolvent
. 81
6.4
Structure of the resolvent kernel
. 84
6.5
The stretched product
. 87
x
Contents
7
Spectral
and Scattering Theory
. 93
7.1
Essential and discrete spectrum
. 93
7.2
Absence of embedded eigenvalues
. 95
7.3
Generalized eigenfunctions
. 102
7.4
Scattering matrix
. 105
7.5
Scattering matrices for the runnel and cylinders
. 114
8
Resonances and Scattering Poles
. 117
8.1
Multiplicities of resonances
. 118
8.2
Structure of the resolvent at a resonance
. 119
8.3
Scattering poles
. 124
8.4
Operator logarithmic residues
. 126
8.5
Half-integer points
. 131
8.6
Coincidence of resonances and scattering poles
. 137
9
Upper Bound for Resonances
. 147
9.1
Resonances and zeros of determinants
. 148
9.2
Singular value estimates
. 151
9.3
Upper bound
. 154
9.4
Estimates on model terms
. 156
10
Selberg
Zeta
Function
. 171
10.1
Relative scattering determinant
. 173
10.2
Regularized traces
. 175
10.3
The resolvent 0-trace calculation
. 183
10.4
Structure of the
zeta
function
. 189
10.5
Order bound
. 196
10.6
Determinant of the Laplacian
. 203
11
Wave Trace and
Poisson
Formula
. 207
1
1.1
Regularized wave trace
. 208
11.2
Model wave kernel
. 209
11.3
Wave 0-trace formula
. 211
11.4
Poisson
formula
. 215
12
Resonance Asymptotics
. 223
12.1
Lower bound on resonances
. 223
12.2
Lower bound near the critical line
. 226
12.3
Weyl formula for the scattering phase
. 229
13
Inverse Spectral Geometry
. 237
13.1
Resonances and the length spectrum
. 238
13.2
Hyperbolic trigonometry
. 239
13.3 Teichmüller
space
. 242
13.4
Finiteness of
isospectral
classes
. 248
Contents xi
14
Patterson-Sullivan Theory
. 259
14.1
A measure on the limit set
. 259
14.2
Ergodicity
. 267
14.3
Hausdorff measure of the limit set
. 274
14.4
The first resonance
. 278
14.5
Prime geodesic theorem
. 284
14.6
Refined asymptotics of the length spectrum
.289
15
Dynamical Approach to the
Zeta
Function
. 297
15.1
Schottky groups
. 298
15.2
Symbolic dynamics
. 300
15.3
Dynamical
zeta
function
. 303
15.4
Growth estimates
. 308
A Appendix
. 315
A.I Entire functions
. 315
A.2 Distributions and Fourier transforms
. 320
A.3 Spectral theory
. 324
A.4 Singular values, traces, and determinants
. 330
A.5 Pseudodifferential operators
. 336
References
. 341
Notation Guide
. 351
Index
. 353 |
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| index_date | 2024-07-02T16:41:11Z |
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| language | English |
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| physical | XI, 355 S. Ill., graph. Darst. |
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| series | PM - Progress in Mathematics |
| series2 | PM - Progress in Mathematics |
| spelling | Borthwick, David Verfasser aut Spectral theory of infinite-volume hyperbolic surfaces David Borthwick Boston [u.a.] Birkhäuser 2007 XI, 355 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier PM - Progress in Mathematics 282 Riemann, Surfaces de Spectre (Mathématiques) Riemann surfaces Spectral theory (Mathematics) Hyperbolische Fläche (DE-588)4735194-9 gnd rswk-swf Spektraltheorie (DE-588)4116561-5 gnd rswk-swf Hyperbolische Fläche (DE-588)4735194-9 s Spektraltheorie (DE-588)4116561-5 s DE-604 PM - Progress in Mathematics 282 (DE-604)BV000004120 282 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2839886&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015466712&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borthwick, David Spectral theory of infinite-volume hyperbolic surfaces PM - Progress in Mathematics Riemann, Surfaces de Spectre (Mathématiques) Riemann surfaces Spectral theory (Mathematics) Hyperbolische Fläche (DE-588)4735194-9 gnd Spektraltheorie (DE-588)4116561-5 gnd |
| subject_GND | (DE-588)4735194-9 (DE-588)4116561-5 |
| title | Spectral theory of infinite-volume hyperbolic surfaces |
| title_auth | Spectral theory of infinite-volume hyperbolic surfaces |
| title_exact_search | Spectral theory of infinite-volume hyperbolic surfaces |
| title_exact_search_txtP | Spectral theory of infinite-volume hyperbolic surfaces |
| title_full | Spectral theory of infinite-volume hyperbolic surfaces David Borthwick |
| title_fullStr | Spectral theory of infinite-volume hyperbolic surfaces David Borthwick |
| title_full_unstemmed | Spectral theory of infinite-volume hyperbolic surfaces David Borthwick |
| title_short | Spectral theory of infinite-volume hyperbolic surfaces |
| title_sort | spectral theory of infinite volume hyperbolic surfaces |
| topic | Riemann, Surfaces de Spectre (Mathématiques) Riemann surfaces Spectral theory (Mathematics) Hyperbolische Fläche (DE-588)4735194-9 gnd Spektraltheorie (DE-588)4116561-5 gnd |
| topic_facet | Riemann, Surfaces de Spectre (Mathématiques) Riemann surfaces Spectral theory (Mathematics) Hyperbolische Fläche Spektraltheorie |
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