The determinant of the Lax-Phillips scattering operator

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 ${\zeta }_B^{\pm }(s,z)$ constructed from the resonances associated to $zI-[(1/2)I\pm B]$. We prove the meromorphic continuation in $s$ of ${\zeta }_B^{\pm }(s,z)$ 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.