Unusual poles of the $\zeta$-functions for some regular singular differential operators

TL;DR

This paper investigates the spectral properties of certain differential operators with regular singularities, revealing unusual pole structures in their associated zeta functions due to irrational powers in resolvent asymptotics.

Contribution

It introduces a detailed analysis of the resolvent's asymptotic expansion for operators with regular singularities, showing the emergence of irrational powers affecting zeta function poles.

Findings

Asymptotic expansion of resolvent includes irrational powers of λ.

Pole structure of zeta and eta functions is affected by singularity-induced powers.

Regular singularities lead to non-standard pole distributions in spectral functions.

Abstract

We consider the resolvent of a system of first order differential operators 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 powers of $\lambda$ which depend on the singularity, and can take even irrational values. The consequences for the pole structure of the corresponding $\zeta$ and $\eta$-functions are also discussed.