A New Example of the Effects of a Singular Background on the Zeta Function

Abstract

To motivate our discussion, we consider a 1+1 dimensional scalar field interacting with a static Coulomb-type background, so that the spectrum of quantum fluctuations is given by a second-order differential operator on a single coordinate r with a singular coefficient proportional to 1/r. We find that the spectral functions of this operator present an interesting behavior: the zeta function has multiple poles in the complex plane; accordingly, logarithms of the proper time appear in the heat-trace expansion. As a consequence, the zeta function does not provide a finite regularization of the effective action. This work extends similar results previously derived in the context of conical singularities.

arXiv comment26 pages

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Cited by in corpus (3)

• Local Neumann semitransparent layers: resummation, pair production and duality
• The spectral $\zeta$-function for quasi-regular Sturm--Liouville operators
• The exotic structure of the spectral $\zeta$-function for the Schrödinger operator with Pöschl--Teller potential