The determinant of the Lax-Phillips scattering operator

TL;DR

This paper introduces a new zeta function based on the Lax-Phillips scattering operator for certain Riemann surfaces, providing a way to define determinants and express classical functions like Selberg's zeta function.

Contribution

It develops a Hurwitz-type zeta function from resonances of the Lax-Phillips operator and relates it to Selberg's zeta function and scattering matrix determinants.

Findings

Defined a meromorphic continuation of the zeta function

Expressed Selberg's zeta function in terms of operator determinants

Connected the scattering matrix determinant to the new zeta function

Abstract

Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $\zeta^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $\zeta^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the operator determinants.