Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains

Abstract

We generalize earlier studies on the Laplacian for a bounded open domain $\Omega \in {\Re }^2$ with connected complement and piecewise smooth boundary. We compare it with the quantum mechanical scattering operator for the exterior of this same domain. Using single layer and double layer potentials we can prove a number of new relations which hold when one chooses {\em independently} Dirichlet or Neumann boundary conditions for the interior and exterior problem. This relation is provided by a very simple set of $\zeta$-functions, which involve the single and double layer potentials. We also provide Krein spectral formulas for all the cases considered and give a numerical algorithm to compute the $\zeta$-function.

22 pages, Postscript, A4

References in corpus (1)

• Spectral Duality for Planar Billiards

Cited by in corpus (4)

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• Remarks on interior transmission eigenvalues, Weyl formula and branching billiards
• Bounds on positive interior transmission eigenvalues
• Billiard Systems in Three Dimensions: The Boundary Integral Equation and the Trace Formula