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.

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

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

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.