Interior Dirichlet eigenvalue problem, exterior Neumann scattering problem, and boundary element method for quantum billiards

Abstract

An explicit expression for the Fredholm determinant $D(E)$ appearing in the boundary element method for two-dimensional quantum billiards is derived in terms of the interior Dirichlet eigenenergies and exterior Neumann scattering phase shifts. Not only in the semiclassical regime but also in all order of \ensuremath{\Elzxh}, $D(E)$ admits the factorization into the interior and exterior contributions, where the former has zeros at interior Dirichlet eigenenergies and the latter at resonances of the Neumann scattering. A new aspect of the Kac's inverse problem is also discussed.