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 seperable locally compact space, let $\mu$ be a Radon measure on $X$ with a full support. Let $(t,x,y)\mapsto p(t,x,y)$ be a strictly positive pointwise consistent $\mu$-heat kernel, and assume that the generator $H_p\ge 0$ of the corresponding self-adjoint contraction semigroup in $L^2(X,\mu )$ induces a regular Dirichlet form. Then, given a function $\Psi :(0,1)\to (0,\infty )$ such that the limit ${\lim }_{t\to 0+}p(t,x,x)\Psi (t)$ exists for all $x\in X$, we prove that for every potential $w:X\to \mathbb{R}$ one has $$\underset{t\to 0+}{\lim }\Psi (t)\mathrm{tr}\left({\mathrm{e}}^{-t{H}_p+w}\right)=\int {\mathrm{e}}^{-w(x)}\underset{t\to 0+}{\lim }p(t,x,x)\Psi (t)d\mu (x)<\infty$$ for the Schrödinger type operator $H_p+w$, provided $w$ satisfies very mild conditions at $\infty$, that are essentially only made to guarantee that the sum of quadratic forms $H_p+w/t$ is self-adjoint and bounded from below for small $t$, and to guarantee that $$\int {\mathrm{e}}^{-w(x)}\underset{t\to 0+}{\lim }p(t,x,x)\Psi (t)d\mu (x)<\infty .$$ 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.

arXiv comment16 pages; comments are welcome!