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. 
Luis Escauriaza, Luca Fanelli, Luis Vega
math.AP