ASYMPTOTIC DEVELOPMENTS OF SLOW PERTURBATIONS OF THE PERIODIC SCHRODINGER OPERATOR

In the semi-classical regime we study the eigenvalues of the operators P(A,phi) = (Dy + A(hy))2 + V(y) + phi(hy), where V is periodic with respect to a lattice GAMMA and phi is bounded with all its derivatives. A(x) is a magnetic potential such that all derivates of non-vanishing order are bounded. We obtain an asymptotic expansion in powers of h of tr f (P(A,phi)) for f is-an-element-of C0infinity(I), where I is an interval disjoint from the essential spectrum. In the case of a simple band we give explicitly the coefficients of this expansion.