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

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.