Determinants of Laplacians on surfaces of finite volume

Abstract

Determinants of the Laplace and other elliptic operators on compact manifolds have been an object of study for many years (see [MP, RS, Vor]). Up until now, however, the theory of determinants has not been extended to non-compact situations, since these typically involve a mixture of discrete and continuous spectra. Recent advances in this theory, which are partially motivated by developments in mathematical physics, have led to a connection, in the compact Riemann surface case, between determinants of Laplacians on spinors and the Selberg zeta function of the underlying surface (see [DP, Kie, Sar, Vor]).

Our purpose in this paper is to introduce a notion of determinants on non-compact (finite volume) Riemann surfaces. These will be associated to the Laplacian Δ shifted by a parameters(1−s), and will be defined in terms of a Dirichlet series ζ(w, s) which is a sum that represents the discrete as well as the continuous spectrum. It will be seen to be regular atw=0, and our main theorem (see Sect. 1) will express exp$\left( {\left. { - \frac{\partial }{{\partial w}}\zeta (w,s)} \right|_{w = 0} } \right)$ as the Selberg zeta function of the surface times the appropriate Γ-factor.