User:DVD206/"Can One Hear the Shape of a Drum?"

Let $$\lambda_k$$ be the Dirichlet eigenvalues for the domain $$\Omega$$, that is, the eigenvalues of the Dirichlet problem for the Laplacian:



\begin{cases} \Delta u + \lambda u = 0, \\ u|_{\partial \Omega} = 0. \end{cases} $$

Since the system above is self-adjoint, the eigenvalues are positive. Two domains are said to be isospectral if they have the same Laplacian eigenvalues.

One can state the discrete analog of the question on graphs...