Heat-kernel expansion on non compact domains and a generalised zeta-function regularisation procedure

TL;DR

This paper explores heat-kernel expansion and zeta function regularisation for Laplace operators on non-compact domains, revealing logarithmic terms and proposing a generalized regularisation method for functional determinants.

Contribution

It introduces a generalized zeta-function regularisation procedure applicable to non-compact domains with complex spectral properties.

Findings

Logarithmic terms appear in heat-kernel expansion for certain non-compact domains.

The zeta function can have higher-order poles at the origin, complicating regularisation.

A new regularisation method is proposed for meaningful evaluation of functional determinants.

Abstract

Heat-kernel expansion and zeta function regularisation are discussed for Laplace type operators with discrete spectrum in non compact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several examples. It is pointed out that for a class of exponential (analytic) interactions, generically the non-compactness of the domain gives rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic continuation of the associated zeta function is investigated. A simple model is considered, for which the analytic continuation of the zeta function is not regular at the origin, displaying a pole of higher order. For a physically meaningful evaluation of the related functional determinant, a generalised zeta function regularisation procedure is proposed.