Decaying modes for the wave equation in the exterior of an obstacle

In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λ_k} of Δ on the geometry of the obstacle ??. In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle ??, both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres—for which the eigenvalues {λ_k} can be determined as roots of special functions—upper and lower bounds for the density of the real {λ_k}, and upper and lower bounds for λ₁, the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on ??. Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono-energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.