For spectra containing a zero mode, a renormalized Green's function derivation yields the sum rule of order 1+1/N, matching Rayleigh-Schrodinger perturbation theory to second order.
Exact sum rules for quantum billiards of arbitrary shape
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On the calculation of exact sum rules of rational order for quantum billiards (spectrum with a null eigenvalue)
For spectra containing a zero mode, a renormalized Green's function derivation yields the sum rule of order 1+1/N, matching Rayleigh-Schrodinger perturbation theory to second order.