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.

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

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

The self-dual and self-symmetric 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

$\mathrm{\Lambda }(G)=\left(\begin{array}{cc}2I& B\\ B^T& \mathrm{\Lambda }+2I\end{array}\right)/(\mathrm{\Lambda }+2I),$

where

$B=\left(\begin{array}{ccccc}-1& 0& 0& \dots & -1\\ -1& -1& 0& \dots & 0\\ 0& ⋮& \ddots & \ddots & ⋮\\ ⋮& ⋮& \ddots & -1& 0\\ 0& 0& \dots & -1& -1\end{array}\right)$

is the circular matrix of first differences.