## Advanced Studies in Pure Mathematics

### Spectral Zeta Functions

André Voros

#### 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).

#### Article information

Dates
Received: 20 January 1991
First available in Project Euclid: 15 August 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1534359133

Digital Object Identifier
doi:10.2969/aspm/02110327

Mathematical Reviews number (MathSciNet)
MR1210795

Zentralblatt MATH identifier
0819.11033

#### Citation

Voros, André. Spectral Zeta Functions. Zeta Functions in Geometry, 327--358, Mathematical Society of Japan, Tokyo, Japan, 1992. doi:10.2969/aspm/02110327. https://projecteuclid.org/euclid.aspm/1534359133