Spectral Zeta Functions

Abstract

This article gives a survey of various generalizations of Riemann’s $\zeta$-function, associated with operator spectra and which may be generically called spectral zeta functions. Areas of application include Riemannian geometry (the spectrum of the Laplacian) and quantum mechanics. We review one example of each class in concrete detail: the Laplacian on a compact surface of constant negative curvature, and the Schrödinger operator on the real line with a homogeneous potential $q^{2M}$ ($M$ a positive integer).