On 2D Inverse Problems/"Can One Hear the Shape of a Drum?"

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Let \sigma_{\Delta(\Omega)} = \{\lambda_k\} be the set of Dirichlet eigenvalues for the domain \Omega. That is, the set of eigenvalues of the following Dirichlet problem for the Laplace operator:


\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 then state the inverse problem of describing the equivalence classes of the isospectral relationship, that is the question in the title of this section, see [K], and its discrete analog for networks:

Let

L(G)=\begin{pmatrix} A & B \\ B^T & C \end{pmatrix}

be the Kirchhoff matrix of a network G w/boundary w/N interior vertices. The submatrix C is semi-positive definite and, therefore, has the singular value decomposition:


C = VDV^{-1}=VDV^T=\sum_{k=1}^N\lambda_k v_k v_k^T

Then the matrix of the Dirichlet-to-Neumann operator can be written as


\Lambda(G) = A - BC^{-1}B^T = A - (BV)D^{-1}(BV)^T = A - \sum_{k=1}^N \frac{Bv_k (Bv_k)^T}{\lambda_k}.

The matrix A contains the information about connections bw/boundary nodes of the network G, the diagonal matrix D has the Laplacian eigenvalues of the network G on its diagonal, and the vectors Bvk are discrete analogs of normal derivatives of the Laplacian eigenvalues of G. The matrix of the Dirichlet-to-Neumann operator is, therefore, completely determined by the these vectors and matrices.

Exercise (*). Prove that the inverse problem with the boundary and spectral data, described above, can be reduced to the inverse problem of Calderon.

The construction in the section "Two spectral theorems" provides a counterexample for uniqueness in an inverse spectral problem.