The wave equation approach to an inverse eigenvalue problem for an arbitrary multiply connected drum in ℝ2 with Robin boundary conditions

Abstract

The spectral function μˆ(t)=∑j=1∞exp(−itμj1/2), where {μj}j=1∞ are the eigenvalues of the two-dimensional negative Laplacian, is studied for small |t| for a variety of domains, where −∞<t<∞ and i=−1. The dependencies of μˆ(t) on the connectivity of a domain and the Robin boundary conditions are analyzed. Particular attention is given to an arbitrary multiply-connected drum in ℝ2 together with Robin boundary conditions on its boundaries.