[Dispersion for the wave equation in the exterior of the torus]{Dispersion for the wave equation in the exterior of the torus in three dimensions}

Abstract

We prove dispersive estimates for the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists for the time harmonic problem, we introduce a related approximate operator for the Dirichlet Laplacian in the exterior of a torus. The approximate operator coincides with the Schr\"odinger operator with a P\"oschl-Teller potential and agrees with the Dirichlet Laplacian to leading order. 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 P\"oschl-Teller, a high-frequency analysis of the approximate operator for the wave equation can be made accurately. The operator for this problem gives a close approximation to the $L^1\rightarrow L^{\infty}$ dispersive estimate at a suitable small distance from the torus for the corresponding exterior wave operator with Dirichlet Laplacian.