The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)

1999 ◽  
pp. 73-141 ◽  
Author(s):  
Cheng-Hung Chang ◽  
Dieter H. Mayer
Keyword(s):  
Zeta Function ◽  
Transfer Operator ◽  
Operator Approach ◽  
Wave Forms ◽  
Selberg’S Zeta Function

scholarly journals Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow

2014 ◽  
Vol 36 (1) ◽  
pp. 142-172 ◽  
Author(s):  
ANKE D. POHL
Keyword(s):  
Zeta Function ◽  
Transfer Operator ◽  
Cusp Forms ◽  
Selberg Zeta Function ◽  
Operator Approach ◽  
Fredholm Determinants ◽  
Triangle Groups ◽  
New Formulation ◽  
Image Position ◽  
Determinants Of Transfer

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families:$$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$In this article we show that the operator families${\mathcal{L}}_{s}^{\pm }$arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for$s\in \mathbb{C}$,$\text{Re}s={\textstyle \frac{1}{2}}$, the operator${\mathcal{L}}_{s}^{+}$(respectively${\mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue$s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.

scholarly journals Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

2011 ◽  
Vol 33 (1) ◽  
pp. 247-283 ◽  
Author(s):  
M. MÖLLER ◽  
A. D. POHL
Keyword(s):  
Zeta Function ◽  
Symbolic Dynamics ◽  
Fredholm Determinant ◽  
Transfer Operator ◽  
Orbit Space ◽  
Cusp Forms ◽  
Operator Family ◽  
Triangle Group ◽  
Selberg Zeta Function ◽  
Triangle Groups

AbstractWe characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover, we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

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