On the geometry of semiclassical limits on Dirichlet spaces

Abstract

This paper is a contribution to semiclassical analysis for abstract Schrödinger type operators on locally compact spaces: Let X be a metrizable separable locally compact space, let μ be a Radon measure on X with a full support. Let (t,x,y)↦p(t,x,y) be a strictly positive pointwise consistent μ-heat kernel, and assume that the generator Hp≥0 of the corresponding self-adjoint contraction semigroup in L2(X,μ) induces a regular Dirichlet form. Then, given a function Ψ:(0,1)→(0,∞) such that the limit limt→0+⁡p(t,x,x)Ψ(t) exists for all x∈X, we prove that for every potential w:X→R one haslimt→0+⁡Ψ(t)tr(e−tHp+w)=∫e−w(x)limt→0+⁡p(t,x,x)Ψ(t)dμ(x)<∞ for the Schrödinger type operator Hp+w, provided w satisfies very mild conditions at ∞, that are essentially only made to guarantee that the sum of quadratic forms Hp+w/t is self-adjoint and bounded from below for small t, and to guarantee that∫e−w(x)limt→0+⁡p(t,x,x)Ψ(t)dμ(x)<∞. The proof is probabilistic and relies on a principle of not feeling the boundary for p(t,x,x). In particular, this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete Riemannian manifolds, and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs.