A Perturbation of a Periodic Schr Odingeroperator by a Modulated Decaying Potentialm

We consider a perturbation of a periodic Schrr odinger operator H 0 by a nonnegative potential V with asymptotics V (x) f x j j jxj 2 q(x); > 0; jxj ! 1 where q is also periodic. The total number of eigenvalues of the operator H() = H 0 +V , > 0, in a gap of H 0 is studied. We discuss a diierence of its asymptotics as ! 1 compared to the case q(x) = 1 Abstract. Let (M;) (M; 0 0) be two measure spaces and let : M 0 ! M be any mapping verifying Fubini's property. From an Hilbertian theorem (which generalizes a theorem of S. Gallot and D. Meyer), we deduce a general criterion to compare the spectra of two semibounded self-adjoint operators T and T 0 acting on the Hilbert spaces L 2 (M) and L 2 (M 0) ResP, under the assumption that they verify Kato's inequality with respect to the mapping