On the calculation of exact sum rules of rational order for quantum billiards (spectrum with a null eigenvalue)

TL;DR

This paper extends the calculation of exact sum rules of rational order for quantum billiards to include spectra with a zero eigenvalue, using renormalization and perturbation theory methods.

Contribution

It introduces a renormalization approach to handle zero modes in sum rule calculations for quantum billiards, aligning perturbative results with direct eigenvalue methods.

Findings

Sum rules expressed in terms of traces up to second order

Agreement between perturbation theory and direct eigenvalue calculations

Extension to spectra with null eigenvalues

Abstract

We generalize the calculation of Ref.~\cite{Amore19B} to the case of a spectrum containing a zero mode. Using a renormalization procedure, we express the sum rules in terms of suitable traces and show that the final expressions, calculated up to second order in perturbation theory agree with the results obtained when working directly with the eigenvalues and using Rayleigh-Schr\"odinger perturbation theory.