On 2D Inverse Problems/ An infinite example

The following construction provides an example of an infinite graph, which Dirichlet-to-Neumann operator satisfies the operator equation in the title of this chapter.



\Lambda(G) = \sqrt{L}. $$

The operator equation reflects the self-duality and self-symmetry of the infinite graph.



Exercise (**). Prove that the Dirichlet-to-Neumann operator of the graph with the natural boundary satisfies the functional equation. (Hint) Use the fact that the operator/matrix is the fixed point of the Schur complement



\Lambda(G) = \begin{pmatrix} 2I & B \\ B^T & \Lambda + 2I \end{pmatrix}/ (\Lambda + 2I), $$

where



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

is the circular matrix of first differences.