A perturbative approach to the spectral zeta functions of strings, drums and quantum billiards

TL;DR

This paper develops a third-order perturbative method to explicitly compute spectral zeta functions and heat kernels for strings, drums, and quantum billiards, enabling analysis of small domain deformations and density variations.

Contribution

It introduces a generalized binomial theorem approach for perturbation expansions of spectral functions, applicable even with spectrum degeneracies, and demonstrates its effectiveness through multiple examples.

Findings

Derived explicit spectral zeta functions and heat kernels to third order.

Successfully applied the method to various geometries and boundary conditions.

Validated the approach by reproducing known results for specific cases.

Abstract

We have obtained an explicit expression for the spectral zeta functions and for the heat kernel of strings, drums and quantum billiards working to third order in perturbation theory, using a generalization of the binomial theorem to operators. The perturbative parameter used in the expansion is either the small deformation of a reference domain (for instance a square), or a small variation of the density around a constant value (in two dimensions both cases can apply). This expansion is well defined even in presence of degenerations of the unperturbed spectrum. We have discussed several examples in one, two and three dimensions, obtaining in some cases the analytic continuation of the series, which we have then used to evaluate the corresponding Casimir energy. For the case of a string with piecewise constant density, subject to different boundary conditions, and of two concentric cylinders of very close radii, we have reproduced results previously published, thus obtaining a useful check of our method.