We investigate the Schro dinger operator H=&2+V acting in L 2 (R n ), n 2, for potentials V that satisfy

We investigate the Schrödinger operator H = −d 2 /dx 2 +(γ/x) sin αx+V , acting in L p (R), 1 ≤ p < ∞, where γ ∈ R \ {0}, α > 0, and V ∈ L 1 (R). For |γ| ≤ 2α/p we show that H does not have positive eigenvalues. For |γ| > 2α/p we show that the set of functions V ∈ L 1 (R), such that H has a positive eigenvalue embedded in the essential spectrum σess(H) = [0, ∞), is a smooth unbounded sub-manifold of L 1 (R) of codimension one. Résumé. On examine l'opérateur de SchrödingerSi |γ| ≤ 2α/p, on montre que H n'a aucune valeur caractéristique positive. Si |γ| > 2α/p, on montre que l'ensemble des fonctions V ∈ L 1 (R), telles que H a une valeur caractéristique positive immergée dans le spectre essentiel σess(H) = [0, ∞), est une sous-variété lisse non-bornée de L 1 (R) de codimensionégaleà un.Proof of Theorem 1.1. For p ≥ 1, α > 0, and γ ∈ R \ {0}, let M p (α, γ) be the set of functions V ∈ L 1 (R) for which the operator H Q,p has a positive eigenvalue. It follows from Theorem 2.1 and Proposition A.1 that

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.

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