The Square of the Dirichlet-to-Neumann map equals minus Laplacian

Abstract

The Dirichlet-to-Neumann maps connect boundary values of harmonic functions. It is an amazing fact that the square of the non-local Dirichlet-to-Neumann map for the uniform conductivity 1 on the unit disc equals minus the local(!) Laplace operator on the boundary circle. To establish a new connection between discrete and continuous Dirichlet-to-Neumann maps and for the approximations I construct a finite and an infinite graphs which Dirichlet-to-Neumann map have the same property: \Lambda^2(1) = - d^2/d \theta^2. The construction gives a new continued fraction identity. It is interesting to consider the geometric and probabilistic (trajectories of the random walk) consequences of this localizing identity unifying discrete and continuous equations for potentials.