Borthwick D. Spectral theory of infinite-area hyperbolic surfaces (Boston, 2007). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаBorthwick D. Spectral theory of infinite-area hyperbolic surfaces. - Boston, Mass.: Birkhäuser, 2007. - 355 p. - (Progress in mathematics; Vol. 256). - ISBN 0-8176-4524-1
 

Место хранения: 013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / 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 eigenfunctions ............................. 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

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 funnel 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
   11.1. 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. Teichmuller space .................................... 242
   13.4. Finiteness of isospectral classes .................... 248

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.1. 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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