Exact sum rules for quantum billiards of arbitrary shape

TL;DR

This paper derives explicit sum rules for the eigenvalues of the Laplacian in arbitrary-shaped 2D domains, enabling finite calculations and validation through examples and numerical comparisons.

Contribution

It provides the first explicit formulas for sum rules of eigenvalues in arbitrary 2D domains, accounting for asymptotic behavior and enabling finite, exact calculations.

Findings

Explicit sum rules derived for arbitrary 2D domains.

Sum rules are finite when using appropriate prescriptions.

Theoretical results validated with numerical examples.

Abstract

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