On the geometry of semiclassical limits on Dirichlet spaces

摘要

This paper is a contribution to semiclassical analysis for abstract Schrodinger type operators on locally compact spaces: Let X be a metrizable separable locally compact space, let mu be a Radon measure on X with a full support. Let (t, x, y) bar right arrow p (t, x, y) be a strictly positive pointwise consistent it-heat kernel, and assume that the generator H-p >= 0 of the corresponding self-adjoint contraction semigroup in L-2 (X, mu) induces a regular Dirichlet form. Then, given a function Psi : (0, 1) -> (0, infinity) such that the limit lim(t -> 0+) p(t, x, x) Psi (t) exists for all x is an element of X, we prove that for every potential w : X -> R one has