Heat kernel estimates and the relative compactness of perturbations by potentials

TL;DR

This paper establishes conditions under which perturbations of certain non-negative operators by potentials are relatively compact, using heat kernel estimates that apply broadly to Riemannian manifolds, weighted manifolds, and graphs.

Contribution

It demonstrates that heat kernel estimates with a control function imply relative compactness of potential perturbations for a wide class of geometric and graph structures.

Findings

Heat kernel estimates hold for Laplace-Beltrami operators on all Riemannian manifolds.

Abstract results on relative compactness of perturbations are derived from these estimates.

Explicit control functions are constructed for weighted manifolds and graphs.

Abstract

We consider a self-adjoint non-negative operator $H$ in a Hilbert space $\mathsf{L}^2(X,{\rm d}\mu)$. We assume that the semigroup $(\mathrm{e}^{-t H})_{t>0}$ is defined by an integral kernel, $p$, which allows an estimate of the form $p(t,x,x)\le F_1(x)F_2(t)$ for all $(x,t)\in X\times\mathbb{R_+}$; we refer to $F_1$ as the \emph{control function}. We show that such an estimate leads to rather satisfying abstract results on relative compactness of perturbations of $H$ by potentials. It came as a surprise to us, however, that such an estimate holds for the Laplace-Beltrami operator on \emph{any} Riemannian manifold. In particular, using a domination principle, one can deduce from the latter fact a very general result on the relative compactness of perturbations by potentials of the Bochner Laplacian associated with a Hermitian bundle $(E, h^E,\nabla^E)$ over an arbitrary Riemannian manifold $(M,g)$; in fact, only quantities of order zero in $g$ enter in the estimates. We extend this result to weighted Riemannian manifolds, where under lower curvature bounds on the $\alpha$-Bakry-\'{E}mery tensor one can construct quite explicit control functions, and to any weighted graph, where the control function is expressed in terms of the vertex weight function.