Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions (Q1363357)

Summary: The spectral function \(\Theta(t)= \sum^\infty_{i=1} \exp(-t\lambda_j)\), where \(\{\lambda_j\}^\infty_{j=1}\) are the eigenvalues of the negative Laplace-Beltrami operator \(-\Delta\), is studied for a compact Riemannian manifold \(\Omega\) of dimension \(k\) with a smooth boundary \(\partial\Omega\), where a finite number of piecewise impedance boundary conditions \(\left({\partial\over\partial n_i}+ \gamma_i\right)u= 0\) on the parts \(\partial\Omega_i\) \((i= 1,\dots, m)\) of the boundary \(\partial\Omega\) can be considered, such that \(\partial\Omega= \bigcup^m_{i=1} \partial\Omega_i\), and \(\gamma_i\) \((i=1,\dots, m)\) are assumed to be smooth functions which are not strictly positive. The underlying problem is to determine the geometry of a compact \(k\)-dimensional smooth Riemannian manifold \(\Omega\) with metric tensor \(g= (g_{\alpha\beta})\), from a complete knowledge of the eigenvalues of the negative Laplace-Beltrami operator \[ -\Delta= -{1\over\sqrt{\text{det }g}} {\partial\over\partial x_\alpha} \left[g^{\alpha\beta} \sqrt{\text{det }g} {\partial\over\partial x_\beta}\right],\;\text{ where } g^{-1}= (g^{\alpha\beta}). \]