Carleman estimates and necessary conditions for the existence of waveguides

Abstract

We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $$({\partial }_t^2+△)u=Vu+W\cdot ({\partial }_t,\nabla )u,$$ and find a necessary quantitative condition on the exponential decay in the spatial-variable of nonzero waveguides solutions which depends on the size of $V$ and $W$ at infinity.

12 pages