On 2D Inverse Problems/Solution of the layered inverse problem

The Dirichlet-to-Neumann operator of a rotation invariant network is diagonal in Fourier coordinates. The direct calculation shows that it's determined by its eigenvalues and they are given by the values of the Stieltjes continued fraction evaluated at the eigenvalues of the minus discrete boundary Laplacian L. That is:

$\Lambda_G = \sqrt{L}\beta(\sqrt{L}),$

where

$L = \begin{pmatrix} 2 & -1 & 0 & \ldots & -1 \\ -1 & 2 & -1 & \ldots & 0 \\ 0 & \vdots & \ddots & \ddots & \vdots \\ 0 & \vdots & -1 & 2 & -1 \\ -1 & 0 & \ldots & -1 & 2 \\ \end{pmatrix}$

or

$\Lambda_G \begin{pmatrix} 1 \\ z \\ z^2 \\ \ldots \\ z^{2n} \end{pmatrix} = (z\beta(z))\circ(i(\sqrt{z}-\frac{1}{\sqrt{z}})) \begin{pmatrix} 1 \\ z \\ z^2 \\ \ldots \\ z^{2n} \end{pmatrix}, z^{2n+1}=1$

Therefore, one can reduce the layered inverse problem to the Pick-Nevanlinna interpolation problem w/the eigenvalues being the data. The conductivities of the network are given then by the coefficients of continued fraction and their reciprocals and can be found by the Euclidean algorithm.

The continuous analog of the inverse problem can be transformed to the inverse problem of Krein for a string, discussed earlier in the book.