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Selberg Zeta Functions and Transfer Operators: An Experimental Approach to Singular Perturbations 1st ed. 2017 [Minkštas viršelis]

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  • Formatas: Paperback / softback, 354 pages, aukštis x plotis: 235x155 mm, weight: 5621 g, 43 Illustrations, color; 28 Illustrations, black and white; XV, 354 p. 71 illus., 43 illus. in color., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2139
  • Išleidimo metai: 14-May-2017
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319512943
  • ISBN-13: 9783319512945
Kitos knygos pagal šią temą:
Selberg Zeta Functions and Transfer Operators: An Experimental Approach to Singular Perturbations 1st ed. 2017Didesnis paveikslėlis
  • Formatas: Paperback / softback, 354 pages, aukštis x plotis: 235x155 mm, weight: 5621 g, 43 Illustrations, color; 28 Illustrations, black and white; XV, 354 p. 71 illus., 43 illus. in color., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2139
  • Išleidimo metai: 14-May-2017
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319512943
  • ISBN-13: 9783319512945
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319512945.html
Raktažodžiai:
This book presents a method for evaluating Selberg zeta functions via transfer operators for the full modular group and its congruence subgroups with characters. Studying zeros of Selberg zeta functions for character deformations allows us to access the discrete spectra and resonances of hyperbolic Laplacians under both singular and non-singular perturbations. Areas in which the theory has not yet been sufficiently developed, such as the spectral theory of transfer operators or the singular perturbation theory of hyperbolic Laplacians, will profit from the numerical experiments discussed in this book. Detailed descriptions of numerical approaches to the spectra and eigenfunctions of transfer operators and to computations of Selberg zeta functions will be of value to researchers active in analysis, while those researchers focusing more on numerical aspects will benefit from discussions of the analytic theory, in particular those concerning the transfer operator method and the spect

ral theory of hyperbolic spaces.

Introduction.-Preliminaries.-The Gamma function and the incomplete Gamma functions.-The Hurwitz Zeta Function and the Lerch Zeta Function.-Computation of the spectra and eigenvectors of large complex matrices.-The hyperbolic Laplace-Beltrami operator.-Transfer operators for the geodesic flow on hyperbolic surfaces.-Numerical results for spectra and traces of the transfer operator for character deformations.-Investigations of Selberg zeta functions under character deformations.-Concluding remarks.-Appendices.-References.-Index of Notations.

Recenzijos

This volume is an interesting contribution to a field that still holds a lots of secrets. It will be of great interest to experts. (Ch. Baxa, Monatshefte für Mathematik, Vol. 198 (2), June, 2022)

What makes this book a unique is that it systematically covers effective computation of the spectral terms of the selberg trace formula, namely the eigenvectors, eigenfunctions and resonances. ... The computation of this book gives us a hint as to what actually occurs with the extremely complicated limit The book is self-contained, covering both the theoretical background and the numerical aspects. (Joshua S. Friedman, Mathematical Reviews, May, 2018)

1 Introduction
1(24)
2 Preliminaries
25(14)
2.1 Complex Numbers
25(2)
2.2 Groups and Geometry
27(5)
2.2.1 Euclidean Geometry
29(3)
2.3 Poincare Section and Poincare Map
32(2)
2.4 Numerical Derivation of a Complex Function
34(1)
2.5 The Newton Method
34(2)
2.6 The Precision of Numbers
36(3)
3 The Gamma Function and the Incomplete Gamma Functions
39(4)
4 The Hurwitz Zeta Function and the Lerch Zeta Function
43(26)
4.1 The Euler-MacLaurin Formula and the Bernoulli Numbers
44(1)
4.2 The Hurwitz Zeta Function
45(8)
4.2.1 Application of the Euler-MacLaurin Formula
47(5)
4.2.2 The Implementation of the Hurwitz Zeta Function for s ε C, z ε Q
52(1)
4.2.3 Test of the Implementation
53(1)
4.3 The Lerch Transcendent and the Lerch Zeta Function
53(16)
4.3.1 Application of the Euler-MacLaurin Formula
58(7)
4.3.2 The Implementation of the Lerch Zeta Function for s ε C, z ε Q and A ε R
65(2)
4.3.3 Test of the Implementation
67(2)
5 Computation of the Spectra and Eigenvectors of Large Complex Matrices
69(18)
5.1 Eigenvalues of a Matrix
70(2)
5.1.1 Schur Decomposition
71(1)
5.2 Complex Givens Rotations
72(3)
5.3 Hessenberg Form
75(1)
5.4 The QR Algorithm for Complex Matrices
76(5)
5.4.1 Deflation
76(1)
5.4.2 The Shifted QR Iteration
77(1)
5.4.3 The Final QR Algorithm
78(3)
5.5 A Verification of the Implementation
81(1)
5.6 Computation of Eigenvectors of Quasi Triangular Matrices
82(5)
5.6.1 Backward Substitution for Non-singular Quasi Triangular Matrices
83(1)
5.6.2 Computation of Eigenvectors
84(3)
6 The Hyperbolic Laplace-Beltrami Operator
87(42)
6.1 The Group PSL(2, R) and Congruence Subgroups
88(3)
6.2 Geodesic Flow on Γ\H
91(2)
6.3 The Spectrum of the Hyperbolic Laplace-Beltrami Operator
93(3)
6.4 Involutions of Maass Wave Forms
96(1)
6.5 The Selberg Trace Formula and the Selberg Zeta Function
97(6)
6.5.1 The Selberg Zeta Function for the Geodesic Flow
103(1)
6.6 Character Deformations for Freely Generated Groups
103(10)
6.6.1 Character Deformation for Γ0 (4)
105(5)
6.6.2 Character Deformation for Γ0 (8)
110(3)
6.7 The Induced Representation Uχ
113(3)
6.8 Period Functions
116(7)
6.9 Computational Methods for Eigenfunctions and Spectra
123(6)
7 Transfer Operators for the Geodesic Flow on Hyperbolic Surfaces
129(66)
7.1 Nuclear Operators
130(1)
7.2 The Transfer Operator as a Sum of Composition Operators
131(2)
7.3 Symbolic Dynamics for the Geodesic Flow
133(3)
7.4 A Transfer Operator for SL(2, Z)
136(19)
7.4.1 Results for the Transfer Operator for (Γ0(n), χ) With n = 1 and x = 1
147(8)
7.5 A Transfer Operator for Finite Index Subgroups Γ of SL(2, Z)
155(4)
7.6 The Transfer Operator for Character Deformations
159(17)
7.6.1 The Transfer Operator £(n)β, ε, χ with the Representation Uχ
160(1)
7.6.2 An Analytic Continuation of the Transfer Operator
161(9)
7.6.3 A Nuclear Representation of the Transfer Operator
170(2)
7.6.4 An Approximation of the Transfer Operator
172(4)
7.7 Symmetries of the Transfer Operator and a Factorization of the Selberg Zeta Function
176(19)
7.7.1 The Transfer Operator £(n)β, ε, χ and the Operators Pκ
177(2)
7.7.2 An Algorithm to Determine the Operators Pκ from the Transfer Operator £(n)β, ε, χ
179(5)
7.7.3 Operators Pκ and Involutions jκ of Maass Wave Forms for Γ0 (n) With χ &quiv; 1
184(2)
7.7.4 Operators Pκ for (Γ0 (4), χ(4)α1 α2) and (Γ0 (8), χ(4)α1 α2 α3)
186(4)
7.7.5 Operators Pκ and the Lewis Equation for (Γ0 (n), χ)
190(5)
8 Numerical Results for Spectra and Traces of the Transfer Operator for Character Deformations
195(36)
8.1 Approximation of the Spectra of Transfer Operators and Numerical Verifications
197(10)
8.2 The Equality of the Spectra of £(n)β, + 1, χ and £(n)β, -1, χ
207(3)
8.3 The Spectra and Traces of the Transfer Operator
210(11)
8.4 The Eigenfunctions of the Transfer Operator and Period Functions
221(10)
9 Investigations of Selberg Zeta Functions Under Character Deformations
231(66)
9.1 Tracking of the Zeros of the Selberg Zeta Function in the β-Plane for α ε [ 0, 1/2]
233(7)
9.2 Numerical Results for the Selberg Zeta Function and Its Zeros
240(12)
9.3 The Zeros Z(4)α-1 of the Selberg Zeta Function Z(4) (β, Χ(4)α)
252(10)
9.4 The Zeros Z(4)α, +1 of the Selberg Zeta Function Z(4) (β, Χα(4))
262(22)
9.5 Results for (Γ0 (8), Χα(8) and (Γ0 (4), Χ(4)α1, α2)
284(3)
9.6 Comparison of Numerical Data to Theoretical Results for (Γ0 (4), Χα(4)) with α &rar; 0
287(10)
10 Concluding Remarks
297(4)
A Computational Aspects of the Transfer Operator for the Kac-Baker Model
301(12)
A.1 A Nuclear Representation of the Transfer Operator for the Kac-Baker Model
302(3)
A.2 A Verification of the Implementation of the Approximation of the Transfer Operator
305(1)
A.3 Numerical Results for the Transfer Operator
306(7)
A.3.1 Concluding Discussion of the Numerical Results
307(6)
B Project Morpheus
313(6)
C The Representatives of Γ0 (4) and Γ0 (8) in SL(2, Z)
319(2)
D The Transfer Operator for (Γ0 (8), Χ(8)α1, α2, α3)
321(10)
E The Zeros and Poles of the Selberg Zeta Function for Arithmetic (Γ0 (4), Χα(4)and Arithmetic (Γ0 (8), Χα(8)
331(12)
References 343(6)
Index of Notations 349