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

Abstract

We obtain optimal L 2 -lower bounds for nonzero solutions to -AV + V*¥ = E¥ in R", n > 2, E s R, where V is a measurable complex-valued potential with V(x) = 0(1x1"') as |x| -»• co, for some i e R. We show that if 36 = max(0, 1 -2e) and exp(T|x|' +< )4 / e L 2 (R") for all T > 0, then ¥ has compact support. This result is new for 0 < £ < 1/2 and generalizes similar results obtained by Meshkov for e = 0, and by Froese, Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof for both e < 0 and e. > 1/2. These L 2 -lower bounds are well known to be optimal for t > 1/2 while for t < 1/2 this last is only known for i = 0 in view of an example of Meshkov. We generalize Meshkov's example for e < 1/2 and thus show that for complex-valued potentials our result is optimal for all e e R.1991 Mathematics subject classification: 35J1O, 35B40, 35B60, 81CO5.

Gaussian decay of harmonic oscillators and related models

“…The potential V in (1.7) is complex-valued, and this is quite likely necessary in order to produce examples with such a decay property. Indeed, this kind of question has a stationary counterpart with a well known manifestation in the examples by Meshkov [18] and Cruz-Sanpedro [8]. Also in that case, the complex nature of the perturbation is essential, as it has been recently proved, at least in part, by Kenig, Silvestre and Wang in [17] and extended to more general elliptic operators by Davey, Kenig and Wang in [9].…”

Sharp Hardy uncertainty principle and gaussian profiles of covariant Schrödinger evolutions

“…The fact that the potential in [10] is complex-valued might have an appealing connection with the examples by Cruz-Sampedro and Meshkov in [4,17] about unique continuation at infinity for stationary Schrödinger equations. In particular, an interesting question is still open, concerning with the possibility or not of providing analogous real-valued examples.…”

On the fractional Landis conjecture

“…The proof in [BK05] is based on the Carleman method. In the spirit of the Carleman method, several extensions have been made in [CS99,Dav14,DZ17,DZ18,LW14], which also take singular drift coefficients and potentials into account.…”