A semi-classical trace formula for Schrödinger operators

Abstract

Let S ℏ =−ℏΔ+ V , with V smooth. If 0< E 2 <lim inf V(x) , the spectrum of S ℏ near E 2 consists (for ℏ small) of finitely-many eigenvalues, λ j (ℏ). We study the asymptotic distribution of these eigenvalues about E 2 as ℏ→0; we obtain semi-classical asymptotics for with , in terms of the periodic classical trajectories on the energy surface . This in turn gives Weyl-type estimates for the counting function . We make a detailed analysis of the case when the flow on B E is periodic.