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We have derived explicit expressions for the sum rules of order one of theeigenvalues of the negative Laplacian on two dimensional domains of arbitraryshape. Taking into account the leading asymptotic behavior of theseeigenvalues, as given from Weyl's law, we show that it is possible to definesum rules that are finite, using different prescriptions. We provide theexplicit expressions and test them on a number of non trivial examples,comparing the exact results with precise numerical results.

Exact sum rules for quantum billiards of arbitrary shape