The Dirichlet-to-Robin Transform

TL;DR

This paper introduces a transformation that converts solutions with Dirichlet boundary conditions into solutions satisfying Robin conditions, enabling exact or approximate Green function constructions for wave, heat, and Schrödinger problems, and deriving spectral expansions.

Contribution

The paper presents a simple transformation method to handle Robin boundary conditions, allowing exact Green function construction and spectral analysis, which was not previously straightforward.

Findings

Exact Green functions for wave, heat, and Schrödinger problems with Robin conditions

Derivation of Gutzwiller-type spectral expansions

Identification of subtle distributional convergence issues

Abstract

A simple transformation converts a solution of a partial differential equation with a Dirichlet boundary condition to a function satisfying a Robin (generalized Neumann) condition. In the simplest cases this observation enables the exact construction of the Green functions for the wave, heat, and Schrodinger problems with a Robin boundary condition. The resulting physical picture is that the field can exchange energy with the boundary, and a delayed reflection from the boundary results. In more general situations the method allows at least approximate and local construction of the appropriate reflected solutions, and hence a "classical path" analysis of the Green functions and the associated spectral information. By this method we solve the wave equation on an interval with one Robin and one Dirichlet endpoint, and thence derive several variants of a Gutzwiller-type expansion for the density of eigenvalues. The variants are consistent except for an interesting subtlety of distributional convergence that affects only the neighborhood of zero in the frequency variable.