A SEMICLASSICAL TRACE FORMULA FOR SCHRODINGER-OPERATORS

Let S(h) = -h-DELTA + V on R(n), with V smooth. If 0 < E2 < lim infV(x), the spectrum of S(h) near E2 consists (for h small) of finitely-many eigenvalues, lambda-j(h). We study the asymptotic distribution of these eigenvalues about E2 as h --> 0; we obtain semi-classical asymptotics for [GRAPHICS] with f-epsilon-C0 infinity, in terms of the periodic classical trajectories on the energy surface B(E) = {(zeta)2 + V(x) = E2}. This in turn gives Weyl-type estimates for the counting function # {j;(square-root lambda-j(h) - E) less-than-or-equal-to ch}. We make a detailed analysis of the case when the flow on B(E) is periodic.