Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Unique continuation at infinity of solutions to Schrödinger equations with complex-valued potentials

J. Cruz-Sampedroa1*

a1 Departamento de Matemáticas, Universidad de Las Américas-Puebla, Cholula, Pue. 72820, Mexico and Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A

We obtain optimal L2-lower bounds for nonzero solutions to – ΔΨ + VΔ = EΨ in Rn, n ≥ 2, E xs2208 R where V is a measurable complex-valued potential with V(x) = 0(|x|-c) as |x|→∞, for some εxs2208 R. We show that if 3δ = max{0, 1 – 2ε} and exp (τ|x|1+δxs2208 L2(Rn)for all τ > 0, then Ψ; has compact support. This result is new for 0 < ε ½ and generalizes similar results obtained by Meshkov for = 0, and by Froese, Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof for both ε≤O and ε≥½. These L2-lower bounds are well known to be optimal for ε ≥ ½ while for ε < ½ this last is only known for ε = O in view of an example of Meshkov. We generalize Meshkov's example for ε< ½ and thus show that for complex-valued potentials our result is optimal for all ε xs2208 R.

(Received March 13 1997)

Footnotes

* Supported in part by CONACyT, Mexico.