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

TL;DR

This paper reveals that the square of the Dirichlet-to-Neumann map for the unit disk equals minus the boundary Laplacian, establishing a novel link between boundary operators and differential operators with implications for discrete and continuous models.

Contribution

It introduces a new connection between discrete and continuous Dirichlet-to-Neumann maps, including a finite and infinite graph construction with this property, and derives a new continued fraction identity.

Findings

The square of the Dirichlet-to-Neumann map equals minus the boundary Laplacian.

Constructed graph models replicate this property, linking discrete and continuous cases.

Derived a new continued fraction identity related to the maps.

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.