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  • Proceedings of the Edinburgh Mathematical Society, Volume 42, Issue 1
  • February 1999, pp. 143-153

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

  • J. Cruz-Sampedro (a1)
  • DOI:
  • Published online: 01 January 2009

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.

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3.R. Carmona and B. Simon , Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, V: lower bounds and path integrals, Comm. Math. Phys. 80 (1981), 5998.

5.P. Deift , W. Hunziker , B. Simon and E. Vock , Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems IV, Comm. Math. Phys. 64 (1978), 134.

6.R. Froese and I. Herbst , Exponential lower bounds to solutions of the Schrödinger equation: lower bounds for the spherical average, Comm. Math. Phys. 92 (1983), 7180.

7.R. Froese , L2-Exponential lower bounds to solutions of the Schrödinger equation, Comm. Math. Phys. 87 (1982), 265286.

12.B. Simon , Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447526.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
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