Carleman estimates and necessary conditions for the existence of waveguides

We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$ 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.

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