Unique continuation at infinity of solutions to Schrodinger equations withcomplex-valued potentials

We obtain optimal L-2-lower bounds for nonzero solutions to -Delta Psi + V Psi = E Psi in R-n, n greater than or equal to 2, E is an element of R, whe re V is a measurable complex-valued potential with V(x) = O(\x\(-l)) as \x\ --> infinity, for some epsilon is an element of R. We show that if 3 delta = max{0, 1 - 2 epsilon} and exp(tau\x\(1+delta)) Psi is an element of L-2( R-n) for all tau > 0, then Psi has compact support. This result is new for 0 < epsilon < 1/2 and generalizes similar results obtained by Meshkov for e psilon = 0, and by Froese, Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Os tenhof for both epsilon less than or equal to 0 and epsilon greater than or equal to 1/2. These L-2-lower bounds are well known to be optimal for epsi lon greater than or equal to 1/2 while for epsilon < 1/2 this last is only known for epsilon = 0 in view of an example of Meshkov. We generalize Meshk ov's example for epsilon < 1/2 and thus show that for complex-valued potent ials our result is optimal for all epsilon is an element of R.