We prove unique continuation principles for solutions of evolution Schrodinger equations with time dependent potentials. These correspond to uncertainly principles of Paley-Wiener type for the Fourier transform. Our results extends to a large class of semi-linear Schrodinger equation.
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.
