Spectral functions of non essentially selfadjoint operators

TL;DR

This paper reviews unusual spectral behaviors, including non-standard powers and negative logs in heat-kernel asymptotics, for selfadjoint extensions of symmetric operators with singular coefficients, highlighting their complex zeta-function structures.

Contribution

It uncovers and discusses novel spectral phenomena in non essentially selfadjoint operators with singular coefficients, expanding understanding of their heat-kernel and zeta-function properties.

Findings

Presence of non-standard powers of t in heat-kernel expansion

Occurrence of negative integer powers of log t

Unusual analytic structure of associated zeta-functions

Abstract

One of the many problems to which J.S. Dowker devoted his attention is the effect of a conical singularity in the base manifold on the behavior of the quantum fields. In particular, he studied the small-$t$ asymptotic expansion of the heat-kernel trace on a cone and its effects on physical quantities, as the Casimir energy. In this article we review some peculiar results found in the last decade, regarding the appearance of non-standard powers of $t$, and even negative integer powers of $\log{t}$, in this asymptotic expansion for the selfadjoint extensions of some symmetric operators with singular coefficients. Similarly, we show that the $\zeta$-function associated to these selfadjoint extensions presents an unusual analytic structure.