| Given a (Dirichlet) eigenfunction of a 2d quantum billiard, the boundary domain count is the number of intersections of the nodal lines with the boundary. An integer sequence can be defined by these numbers, sorted according to the energy of the eigenfunction. We demonstrate that these sequences store information about the periodic orbits of the underlying classical problem, in a manner similar to the well known trace formulas of the spectral counting function. For one integrable problem, we describe the analytical derivation of the formulas, involving the inversion of the spectral counting function. For a chaotic billiard, we give numerical evidence for similar behavior. |
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