In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schrodinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
We study via Carleman estimates the sharpest possible exponential decay for {it waveguide} solutions to the Laplace equation $$(partial^2_t+triangle)u=Vu+Wcdot(partial_t, abla)u,$$ 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.
