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

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