On the resolvent and spectral functions of a second order differential operator with a regular singularity

TL;DR

This paper investigates the resolvent and spectral functions of a second order differential operator with a regular singularity, revealing unusual asymptotic behaviors and implications for spectral invariants like the zeta function and heat kernel.

Contribution

It provides a detailed analysis of the resolvent's asymptotic expansion for operators with regular singularities, highlighting novel power behaviors depending on the singularity.

Findings

Unusual powers of λ in the resolvent asymptotics

Altered pole structure of the ζ-function

Modified small-t asymptotics of the heat kernel

Abstract

We consider the resolvent of a second order differential operator with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents unusual powers of $\lambda$ which depend on the singularity. The consequences for the pole structure of the $\zeta$-function, and the small-$t$ asymptotic expansion of the heat-kernel, are also discussed.