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

Published online by Cambridge University Press:  20 January 2009

J. Cruz-Sampedro
J. Cruz-Sampedro
Affiliation:
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
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We obtain optimal L2-lower bounds for nonzero solutions to – ΔΨ + VΔ = EΨ in Rn, n ≥ 2, E ∈ R where V is a measurable complex-valued potential with V(x) = 0(|x|-c) as |x|→∞, for some ε∈ R. We show that if 3δ = max{0, 1 – 2ε} and exp (τ|x|1+δ)Ψ ∈ 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 ∈ R.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 1 , February 1999 , pp. 143 - 153
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

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