On the Singularities of the Zeta and Eta functions of an Elliptic Operator

TL;DR

This paper constructs explicit examples of perturbations in elliptic operators that induce singularities in their zeta and eta functions at all permissible points, revealing generic behavior of these functions.

Contribution

It provides explicit constructions and genericity results for singularities of zeta and eta functions of elliptic operators, including a new proof for Dirac-type operators.

Findings

Explicit examples of perturbations causing singularities at all allowed points.

Genericity results for singularities in various classes of elliptic operators.

A new proof of a known result for Dirac-type operators.

Abstract

Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equal to n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions be singular at all the points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, selfadjoint first-order differential operators, and selfadjoint elliptic pseudodifferential operators. As a result, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a new proof of a well known result of Branson-Gilkey, which was obtained by means of Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of the third author's paper [Po1]. Corrections to that statement are given in Appendix B.