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2017 | Buch

Kapitel lesen Erstes Kapitel lesen

Selberg Zeta Functions and Transfer Operators

An Experimental Approach to Singular Perturbations

verfasst von: Markus Szymon Fraczek

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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 spectral theory of hyperbolic spaces.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In recent years the application of the transfer operator method in the study of Selberg zeta functions and the spectral theory of hyperbolic spaces has made significant progress, in both analytical investigations and numerical investigations. We consider transfer operators for the geodesic flow on surfaces of constant negative curvature, therefore systems where a particle is moving freely on such a surface with constant velocity.

Markus Szymon Fraczek

Chapter 2. Preliminaries

Abstract

In this chapter we will recall some basic facts, which we will need later on.

Markus Szymon Fraczek

Chapter 3. The Gamma Function and the Incomplete Gamma Functions

Abstract

The gamma function is defined for $s \in \mathbb{C}$ by

$$\displaystyle{ \varGamma \left (s\right ) =\int _{ 0}^{\infty }t^{s-1}e^{-t}dt }$$

Markus Szymon Fraczek

Chapter 4. The Hurwitz Zeta Function and the Lerch Zeta Function

Abstract

In this chapter we will discuss formulas we have developed for the evaluation of certain zeta functions. We will need them later for the numerical computation of the spectrum of the transfer operator. The implementations of these zeta functions are in a sense the heart of our computations, so we need to be very careful. Unfortunately we have found approximations of these zeta functions in the literature which are quite limited; this is the case especially for the Lerch zeta function. This is understandable, since these zeta functions are special functions, which compared to others are not used so often.

Markus Szymon Fraczek

Chapter 5. Computation of the Spectra and Eigenvectors of Large Complex Matrices

Abstract

One of the critical points in our numerical investigation of the transfer operator is the computation of its eigenvalues. In this section we want to describe briefly what problems arise when computing the eigenvalues of the transfer operator and how we can overcome these problems. To get the best results, both with respect to accuracy and computation time, we had to combine several techniques to produce an optimal algorithm.

Markus Szymon Fraczek

Chapter 6. The Hyperbolic Laplace-Beltrami Operator

Abstract

In this chapter we will introduce some basic concepts of hyperbolic geometry and automorphic forms. A variety of books is available which provide a more comprehensive description of the relevant material. Hejhal’s books about the Selberg trace formula [58] and [59] are a source of exhaustive informations regarding most topics discussed in this chapter, these books are most useful for researches already familiar with most of the concepts. Iwaniec’s book [68] is more introductory in nature, discussing the relevant subjects in an accessible way. Bump’s book [25] covers both the classical and the representation theoretic views of automorphic forms. Bruggeman’s book on families of automorphic forms [21] is especially relevant in regard of deformations of automorphic forms, discussing their dependency on the weight and the character. For introductory articles on the spectral theory on hyperbolic surfaces and the Selberg trace formula see [14] and [83].

Markus Szymon Fraczek

Chapter 7. Transfer Operators for the Geodesic Flow on Hyperbolic Surfaces

Abstract

In this chapter we will discuss a transfer operator for the geodesic flow on hyperbolic surfaces, which Fredholm determinant gives the Selberg zeta function. The transfer operators we are interested in are a so-called nuclear operators of order zero, these operators have a well defined trace and can be approximated by operators of finite rank. Transfer operators can be also regarded as composition operators, for which an explicit trace formula can be found.

Markus Szymon Fraczek

Chapter 8. Numerical Results for Spectra and Traces of the Transfer Operator for Character Deformations

Abstract

In Chap. 7 we discussed how to evaluate the Selberg zeta function $Z^{\left (n\right )}(\beta,\chi )$ by computing the spectrum of the transfer operators

$$\displaystyle{\tilde{\mathcal{L}}_{\beta,\chi }^{\left (n\right )} = \left (\begin{array}{cc} 0 &\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )} \\ \mathcal{L}_{\beta,-1,\chi }^{\left (n\right )} & 0 \end{array} \right ),\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}\mathcal{L}_{\beta,-1,\chi }^{\left (n\right )}\text{ and }\mathcal{P}_{ k}\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}.}$$

To obtain a numerical approximation of the spectrum of the transfer operator $\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}$ in Proposition 7.5

$$\displaystyle\begin{array}{rcl} \left [\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}\vec{f}\left (z\right )\right ]_{ i}& =& \sum _{k=0}^{\infty }\sum _{ s=0}^{\infty }\sum _{ m=1}^{n}\sum _{ j=1}^{\mu _{n} }\left [U^{\chi }\left (ST^{m\varepsilon }\right )\right ]_{ i,j}\frac{f_{j}^{\left (k\right )}\left (1\right )} {k!} \sum _{t=0}^{k}\binom{k}{t}\frac{\left (-1\right )^{k-t+s}} {n^{2\beta +t+s}} {}\\ & & \frac{1} {s!} \frac{\varGamma (2\beta +t+s)} {\varGamma (2\beta + t)} \varPhi \!\left (\chi \left (r_{j}^{\left (n\right )}T^{n\varepsilon }\left (r_{ j}^{\left (n\right )}\right )^{-1}\right ),2\beta +t+s, \frac{m+1} {n} \right )\!\left (z-1\right )^{s}, {}\\ \end{array}$$

we approximate this operator by the matrix $\mathcal{M}_{\beta,\varepsilon,\chi }^{\left (n\right ),N}$ in Proposition 7.7

$$\displaystyle\begin{array}{rcl} \left [\left (\mathcal{M}_{\beta,\varepsilon,\chi }^{\left (n\right ),N}\right )_{ s,k}\right ]_{i,j}& =& \frac{1} {s!}\sum _{t=0}^{k}\binom{k}{t}\frac{\left (-1\right )^{k-t+s}} {n^{2\beta +t+s}} \frac{\varGamma (2\beta + t + s)} {\varGamma (2\beta + t)} \sum _{m=1}^{n}\left [U^{\chi }\left (ST^{m\varepsilon }\right )\right ]_{ i,j} {}\\ & & \varPhi \left (\chi \left (r_{j}^{\left (n\right )}T^{n\varepsilon }\left (r_{ j}^{\left (n\right )}\right )^{-1}\right ),2\beta + t + s, \frac{m + 1} {n} \right ) {}\\ \end{array}$$

and compute its spectrum.

Markus Szymon Fraczek

Chapter 9. Investigations of Selberg Zeta Functions Under Character Deformations

Abstract

In this chapter we present the numerical results we obtained using our computer program package Morpheus for the transfer operators $\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}$ in Sect. 7.6 and the Selberg zeta function $Z^{\left (n\right )}(\beta,\chi )$ for $\varGamma _{0}\left (n\right )$ and character χ given by

$$\displaystyle{Z^{\left (n\right )}(\beta,\chi ) =\det \left (1 -\tilde{\mathcal{L}}_{\beta,\chi }^{\left (n\right )}\right ) =\det \left (1 -\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}\mathcal{L}_{\beta,-1,\chi }^{\left (n\right )}\right ) =\det \left (1 -\mathcal{L}_{\beta,-1,\chi }^{\left (n\right )}\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}\right ).}$$

Markus Szymon Fraczek

Chapter 10. Concluding Remarks

Abstract

The results in the previous chapters describe several phenomena for the transfer operator and the Selberg zeta function $Z^{\left (n\right )}(\beta,\chi )$ which seem thus far to be unknown even to the experts. Clearly, one of our basic results is the symmetries of the transfer operator in Sect. 7.7 whose existence we found by investigating a new form of the transfer operator that we derived in Lemma 7.1. This form allows us to write down explicitly the action of the transfer operator on every component of a vector-valued function for any given group $\varGamma _{0}\left (n\right )$ with a character χ, (see, e.g., Appendix D for the form of the transfer operator for $\varGamma _{0}\left (8\right )$).

Markus Szymon Fraczek

Backmatter

Titel: Selberg Zeta Functions and Transfer Operators
verfasst von: Markus Szymon Fraczek
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-51296-9
Print ISBN: 978-3-319-51294-5
DOI: https://doi.org/10.1007/978-3-319-51296-9