Discrete spectrum of an elliptic periodic differential operator perturbed by a differential operator with decaying coefficient

We study the discrete spectrum arising in the spectral gaps of an elliptic periodic differential operator; P, perturbed by a differential operator, Q, with decaying coefficient. In the semi-classical regime (Q = Q(h), h SE arrow 0) (resp. the large-coupling constant: Q = lambda Q, lambda --> +infinity), we obtain an asymptotic expansion in powers of IL (resp. lambda) of trf(P + Q(h)) (resp. trf(P + lambda Q)) and we give explicitly the lending term. Here f is an element of C-o(infinity)(I) where I is an interval disjoint from the spectrum of P. We apply these results to study the asymptotic distribution of eigenvalues. (C) Academie des Sciences/Elsevier, Paris.