Quantum chaos and mesoscopic systems: mathematical methods in the quantum signatures of chaos
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
1997
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| Series: | Mathematics and its applications
397 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 331 S. |
| ISBN: | 0792344596 |
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MARC
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| 245 | 1 | 0 | |a Quantum chaos and mesoscopic systems |b mathematical methods in the quantum signatures of chaos |c by Norman E. Hurt |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 1997 | |
| 300 | |a XV, 331 S. | ||
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Record in the Search Index
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| adam_text | Table of Contents
Preface xiii
1 Signatures of Quantum Chaos 1
1.1 Introduction 1
1.2 Spectral Staircase 5
1.3 Unfolding the Spectrum 6
1.4 Hyperbolic Triangles: An Example 6
1.4.1 Artin s Billiards 7
1.4.2 Quaternion Algebras 8
1.4.3 Arithmetic Groups 8
1.4.4 Tiling Triangles 9
1.4.5 Hecke Triangles 11
1.5 Polygonal Billiards 11
1.6 The Cardioid Billiard 11
1.7 The Oval 11
1.7.1 Circle Problem 13
1.8 Torus 15
1.9 Surface of Revolution 15
1.10 Liouville Surface 17
1.11 Scaling and Transition for Integrable Systems 17
1.12 Zoll Surface 21
1.13 Random Matrix Theory 21
1.14 Short Range Correlation 22
1.15 Integrable and Chaotic Systems: RMT Conjectures 23
1.16 Number Variance 23
1.17 Spectral Rigidity and Saturation 25
1.18 Spectral Form Factor 25
1.19 Exact Spectral Form Factor Theorem 26
1.20 Berry s Semiclassical Theorem 27
1.20.1 Degeneracy of Orbits 27
1.20.2 Democracy: Classical Sum Rule 27
1.20.3 Berry s Trick 27
1.21 Example: Rectangular Billiards 29
1.22 Saturation for Integrable Systems 30
1.23 Saturation Values for GOE and GUE: Semiclassical Results 31
1.24 Gaussian Fluctuation in RMT 31
1.25 Selberg Trace Formula 32
1.26 Gutzwiller s Trace Formula 33
1.27 Gutzwiller for Plane Billiards 33
1.28 Bolte s Semiclassical Statistics 34
1.29 Selberg Trace Formula for Hyperbolic Plane Billiards .... 35
1.29.1 Selberg Zeta Function 36
1.29.2 Artin s Billiards and Venkov Zograf Factorization . . 36
1.29.3 Huber s Law 37
1.29.4 Mean Multiplicity 37
1.30 Riemann Zeta Function 37
1.31 Mode Fluctuation Distribution 41
1.32 RMT Classes Revisited 42
1.33 Triangles Da Capo 45
1.34 Montgomery Dyson Hypothesis 45
1.35 i Functions 46
1.35.1 L—Functions Encore 48
1.36 Selberg s Moment Theorem for L Functions 49
1.37 Dyson s Autocorrelation Conjecture 50
1.38 N Level Correlation: Semiclassical Calculations 50
1.39 The Hardy Littlewood Conjecture 51
1.40 Zeros of Principal L Functions 52
1.41 Modular Billiards: Two Point Correlation Form Factor ... 53
1.42 Geometrically Finite Spaces 53
1.42.1 Exponent of Convergence 53
1.42.2 Lattice Point Problem 55
1.43 Geometric Structure 56
1.44 STF for Geometrically Finite Spaces 56
1.45 Length Spectra for Hyperbolic Surfaces 59
1.46 Hyperbolic 3 Orbifold 59
1.47 Slow Oscillations 62
1.48 Chaos in Electronic Band Structure 63
1.49 Magnetization and Susceptibility 63
1.50 Gutzwiller Scattering Model 65
1.51 Experimental Work 66
1.51.1 Microwave Cavities 66
1.51.2 Mesoscopic Devices 69
2 Billiards: Polygonal and Others 71
2.1 Introduction 71
2.2 Rational Billiards 71
2.3 Entropy and Mixing 72
vi
2.4 Billiard Groups 72
2.5 Billiard Motion 72
2.6 Periodic Orbits 73
2.7 Regular Polygons and Zeta Function 75
2.8 Veech Polygons 76
2.9 Fermat Curves 77
2.10 Monodromy Map 77
2.11 Numerical Results: Quantum Billiards 78
2.11.1 Rational and Irrational Billiards 78
2.11.2 Staircase Billiards 78
2.11.3 Pure Rhombus Billiard 79
2.12 II/4 Right Triangles 79
2.13 Richens Truncated Triangle 80
2.14 GWW Models 81
2.15 Pseudo integrable L shaped Billiard 81
2.16 Length Spectra for Pseudo integrable Billiards 82
2.17 Sinai Billiards 83
2.18 Point Sinai Billiard 84
2.19 Bunimovich Stadium 85
2.20 Spectral Autocorrelation and Survival Probability 85
2.21 Cardioid Billiard 86
2.22 Hyperbola 88
3 Quantum Transition Amplitudes 89
3.1 Introduction 89
3.2 Distributions of Matrix Elements 95
3.3 Quantum Ergodic Systems 99
3.4 Random Eigenfunctions 101
3.5 Trapping 104
3.6 Coulombic Periodic Potentials 106
3.7 Quantized Hyperbolic Toral Automorphisms 107
3.8 Correlations Ill
3.9 Hyperbolic Toral Automorphisms 112
3.10 Equidistribution Results 115
3.11 Prime Geodesic Theorem 116
3.12 Billiards Flow 119
3.13 Rate of Quantum Ergodicity 119
3.14 Ratner s Central Limit Theorem 120
3.15 Recent Results on Tori 121
3.16 Trace Formula for the Quantized Cat Map 121
3.17 Appendix 122
vii
4 Variance of Quantum Matrix Elements 125
4.1 Introduction 125
4.2 Variance of Quantum Matrix Elements 125
4.3 Berry s Trick and the Hyperbolic Case 126
4.4 Nonhyperbolic Case 128
4.5 Random Matrix Theory 128
4.6 Baker s Map and Other Systems 129
4.7 Appendix: Baker s Map 129
5 Error Terms 133
5.1 Introduction 133
5.2 The Riemann Zeta Function in Periodic Orbit Theory . . . 135
5.3 Form Factor for Primes 137
5.4 Error Terms in Periodic Orbit Theory: Co compact Case . . 138
5.5 Binary Quadratic Forms as a Model 139
6 Co Finite Model for Quantum Chaology 141
6.1 Introduction 141
6.2 Co finite Models 141
6.3 Geodesic Triangle Spaces 144
6.4 L Functions 145
6.5 Zelditch s Prime Geodesic Theorem 146
6.6 Zelditch s Pseudo Differential Operators 147
6.7 Weyl s Law Generalized 148
6.8 Equidistribution Theory 150
7 Landau Levels and L Functions 153
7.1 Introduction 153
7.2 Landau Model: Mechanics on the Plane and Sphere 153
7.3 Landau Model: Mechanics on the Half Plane 155
7.4 Selberg s Spectral Theorem 157
7.5 Pseudo Billiards 158
7.6 Landau Levels on a Compact Riemann Surface 159
7.7 Automorphic Forms 160
7.8 Maass Selberg Trace Formula 162
7.9 Degeneracy by Selberg 163
7.10 Hecke Operators 163
7.11 Selberg Trace Formula for Hecke Operators 167
7.12 Eigenvalue Statistics on X 169
7.13 Mesoscopic Devices 170
7.14 Hall Conductance on Leaky Tori 170
7.15 L Functions, One More Time 171
7.16 Maass Cusp Forms 173
viii
7.17 Equidistribution and Quantum Ergodicity 175
7.18 Alternative Zeta Functions 177
7.19 Infinite Volume Case 178
8 Wigner Time Delay 179
8.1 Introduction 179
8.2 Gutzwiller Model 179
8.2.1 Example: Artin Surface 179
8.3 Time Delay Function 180
8.3.1 Example: Artin Surface 180
8.3.2 Example: Gutzwiller Model 181
8.4 Phase Shift Asymptotics 182
8.4.1 Example: Gutzwiller Model 182
8.5 Resonances and Poles of the Scattering Matrix 183
8.6 Density of Riemann Zeros 183
8.7 Correlation Function of r(k) 184
8.8 Gutzwiller Model in a Magnetic Field 186
8.9 Winding Number 187
8.10 Miiller s Admissable Surfaces 188
8.11 Scattering Determinants for Congruence Groups 190
8.12 Semiclassical Expansion 191
8.13 Semiclassical Results for the Wigner Time Delay 193
8.14 Appendix 195
9 Scattering Theory for Leaky Tori 197
9.1 Introduction 197
9.2 Miiller s Admissable Surfaces 197
9.3 Scattering Operators 199
9.4 Weyl s Law for Mesoscopic Systems 201
9.5 Miiller s Trace Formula 203
9.6 Scattering Theory on Hyperbolic Half Cylinders 203
9.7 Hyperbolic Half Cylinders 204
9.8 Poschl Teller Hamiltonians 204
9.9 Scattering Theory on Hyperbolic Half Cylinders 205
9.10 Scattering Theory for Two Strictly Convex Bodies 206
9.11 Diffraction and Resonances 207
10 Dissolving Bound States 211
10.1 Introduction 211
10.2 Quantum Mechanics on Leaky Tori 212
10.3 Eisenstein Series and Scattering Matrices 214
10.3.1 Modular Group 215
10.3.2 Gutzwiller s Leaky Tori 216
ix
10.4 Congruence Subgroups 216
10.5 Lattice Deformations 217
10.6 Fermi Golden Rule 219
10.7 Essentially Cuspidal 220
10.8 Deformation of Character 223
10.9 Length Spectra of Mesoscopic Systems 225
10. lOUpper Bounds on the Number of Resonances 226
10.11 Conclusions 226
11 Dissolving Eigenvalues 227
11.1 Introduction 227
11.2 The Bottom of the Continuous Spectrum 229
11.3 Dissolving Degenerate Eigenvalues 230
11.4 Small Eigenvalues 231
12 Half Integral Forms 235
12.1 Introduction 235
12.2 The Shimura Correspondence 236
12.2.1 Fourier Coefficients 237
12.2.2 Iwaniec s Estimate for Fourier Coefficients of Half
Integral Forms 238
12.3 Shintani s Map 239
12.3.1 Kohnen Zagier Example Again 240
12.4 Maass Forms 241
12.5 Maass Forms of Half Integral Weight 242
12.6 Shimura s Correspondence for Maass Forms 243
12.7 Spectra of Landau States 244
12.8 Fourier Coefficients of Maass Forms 245
12.9 Distribution of Closed Geodesies on PSL(2,R) H 246
12.10Conclusion 246
12.11Appendix 247
12.11.1 Theta Series 247
12.11.2 Niwa s Construction 249
12.11.3GeneralShintaniMap 249
12.11.4Newforms and Oldforms 251
12.11.5 Number of Inequivalent Cusps 252
13 Isometric and Isospectral Manifolds 253
13.1 Introduction 253
13.2 Lattices and Spectra 254
13.3 Hyperbolic Spaces 254
13.4 Isospectral Deformations 255
13.5 Sunada s and Board s Theorems 255
x
13.6 Schrodinger Operators 256
13.7 Heisenberg Manifolds 256
13.8 Spectra of Heisenberg Manifolds 257
13.9 Length Spectra of Heisenberg Manifolds 257
13.10Poisson Formula for Heisenberg Manifolds 258
13.11Two Problems of Zelditch 259
13.12Sarnak s Conjecture 261
13.13Appendix 261
14 Mesoscopic Structures 263
14.1 Introduction 263
14.2 Scattering Matrix 266
14.3 Probability Distribution of the A s and Level Repulsion . . 267
14.4 Eigenvalue Statistics 268
14.5 Dyson Mehta Formula and Beenakker s Generalization . . . 269
14.6 Dyson Beenakker Integral Equation 269
14.7 Beenakker s Variance Formula 270
14.8 Applications of the Dyson Mehta Beenakker Formula .... 271
14.8.1 Conductance 271
14.8.2 Shot Noise Power 272
14.8.3 Suppression of Shot Noise Power 272
14.8.4 Normal Superconductor Interface 273
14.8.5 Quantum Point Contact 274
14.8.6 Josephson Junction 274
14.9 Diffusion Equation Approach 274
14.9.1 Inelastic Scattering 276
14.10Disordered Metallic Wires 276
14.11Dyson s Large iV Expansion 278
14.12Universality of Weak Localization 278
14.13Applications to Quantum Dots 279
14.13.1 Probability Distribution of the A s 279
14.13.2 Conductance of a Quantum Dot 281
14.13.3 Ballistic Shot Noise for Quantum Dot 281
14.13.4 Normal Superconductor Interface 282
14.14Small iV Results 282
14.15Conductance Distribution for Quantum Dot 283
14.16Resonance Statistics 284
14.16.1 Mesoscopic Resonance Width 284
14.16.2 Microwave Cavity Resonance Statistics 285
14.17Disordered Metals 285
14.18Parametric Correlations 286
14.19Semiclassical Results 289
xi
14.20Quantum Hall Effect 291
14.21Testing for Chaos 292
14.22Quantum Point Contacts: In Plane Gate Devices 293
14.23Some Concluding Remarks 293
14.24Appendix 295
15 References 297
Index 329
xii
|
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| id | DE-604.BV011577294 |
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| indexdate | 2024-07-09T18:12:08Z |
| institution | BVB |
| isbn | 0792344596 |
| language | English |
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| series2 | Mathematics and its applications |
| spelling | Hurt, Norman E. Verfasser aut Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos by Norman E. Hurt Dordrecht [u.a.] Kluwer Acad. Publ. 1997 XV, 331 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 397 Chaos (théorie des systèmes) ram Mathematische Physik Mathematical physics Mesoscopic phenomena (Physics) Quantum chaos Mesoskopisches System (DE-588)4280799-2 gnd rswk-swf Quantenchaos (DE-588)4130849-9 gnd rswk-swf Quantenchaos (DE-588)4130849-9 s Mesoskopisches System (DE-588)4280799-2 s DE-604 Mathematics and its applications 397 (DE-604)BV008163334 397 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007796249&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hurt, Norman E. Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos Mathematics and its applications Chaos (théorie des systèmes) ram Mathematische Physik Mathematical physics Mesoscopic phenomena (Physics) Quantum chaos Mesoskopisches System (DE-588)4280799-2 gnd Quantenchaos (DE-588)4130849-9 gnd |
| subject_GND | (DE-588)4280799-2 (DE-588)4130849-9 |
| title | Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos |
| title_auth | Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos |
| title_exact_search | Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos |
| title_full | Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos by Norman E. Hurt |
| title_fullStr | Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos by Norman E. Hurt |
| title_full_unstemmed | Quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos by Norman E. Hurt |
| title_short | Quantum chaos and mesoscopic systems |
| title_sort | quantum chaos and mesoscopic systems mathematical methods in the quantum signatures of chaos |
| title_sub | mathematical methods in the quantum signatures of chaos |
| topic | Chaos (théorie des systèmes) ram Mathematische Physik Mathematical physics Mesoscopic phenomena (Physics) Quantum chaos Mesoskopisches System (DE-588)4280799-2 gnd Quantenchaos (DE-588)4130849-9 gnd |
| topic_facet | Chaos (théorie des systèmes) Mathematische Physik Mathematical physics Mesoscopic phenomena (Physics) Quantum chaos Mesoskopisches System Quantenchaos |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007796249&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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