AN INVERSE PROBLEM FOR A GENERAL DOUBLY-CONNECTED BOUNDED DOMAIN: AN EXTENSION TO HIGHER DIMENSIONS

The spectral function $\Theta(t)=\sum_{\nu=1}^\infty \exp(-t\lambda_\nu)$, where $\{\lambda_\nu\}_{\nu=1}^\infty$ are the eigenvalues of the negative Laplacian $-\nabla^2=-\sum_{i=1}^3(\frac{\partial}{\partial x_i})^2$ in the $(x^1, x^2, x^3)$-space, is studied for an arbitrary doubly connected bounded domain $\Omega$ in $R^3$ together with its smooth inner bounding surface $\tilde S_1$ and its smooth outer bounding surface $\tilde S_2$, where piecewise smooth impedance boundary conditions on the parts $S_1^*$, $S_2^*$ of $\tilde S_1$ and $S_3^*$, $S_4^*$ of $\tilde S_2$ are considered, such that $\tilde S_1=S_1^*\cup S_2^*$ and $\tilde S_2=S_3^*\cup S_4^*$.