An inverse eigenvalue problem for an arbitrary, multiply connected, bounded domain in R 3 with impedance boundary conditions

The basic problem in this paper is that of determining the geometry of an arbitrary, multiply connected, bounded region in $R^3 $, together with the impedance boundary conditions, from a complete knowledge of the eigenvalues $\{ \lambda _j \}_{j = 1}^\infty $ for the negative Laplacian $ - \nabla ^2 = - \sum_{i = 1}^3 (\partial/\partial x^i )^2$ in the $(x^1 ,x^2 ,x^3 )$-space, using the asymptotic expansion of the spectral function $\Theta (t) = \sum\nolimits_{j = 1}^\infty \exp ( - t\lambda _j )$ for small positive t.

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