The Dirichlet-to-Robin transform

A simple transformation converts a function satisfying a Dirichlet boundary condition to a function satisfying a Robin (generalized Neumann) condition. Under sufficiently restrictive conditions, one can arrange that the old function is the solution of a solvable differential equation and the new one solves an interesting differential equation. 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 neighbourhood of zero in the frequency variable.