A Pöschl--Teller approximation to the wave equation outside a torus

Abstract

We prove dispersive estimates for an approximation to the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists in toroidal coordinates for the time harmonic problem, we introduce a related approximate operator for the Dirichlet Laplacian in the exterior of a torus. The approximate operator separates into Schrödinger operators with Pöschl-Teller potentials after freezing $N=N_0$ in the conjugated Laplacian. The operator here which we develop is related to the so-called Mehler-Fock kernel. Using the known solution to the eigenvector and eigenvalue problem of Pöschl-Teller, a high-frequency analysis of the approximate operator for the wave equation can be made accurately. The operator for this problem gives an angularly truncated $L^1\to L^{\infty }$ dispersive estimate on compact coordinate collars. Duhamel's principle controls its error for the corresponding exterior wave operator with Dirichlet Laplacian.