Semiclassical limits of quantum partition functions on infinite graphs

Abstract

We prove that if $H$ denotes the operator corresponding to the canonical Dirichlet form on a possibly locally infinite weighted graph $(X,b,m)$, and if $v:X\to \mathbb{R}$ is such that $H+v/\hslash$ is well-defined as a form sum for all $\hslash >0$, then the quantum partition function $\mathrm{tr}({\mathrm{e}}^{-\beta \hslash (H+v/\hslash )})$ satisfies $$\mathrm{tr}({\mathrm{e}}^{-\beta \hslash (H+v/\hslash )})\stackrel{\hslash \to 0+}{\to }\sum _{x\in X}{\mathrm{e}}^{-\beta v(x)}\text{ for all β>0},$$ regardless of the fact whether ${\mathrm{e}}^{-\beta v}$ is apriori summable or not. We also prove natural generalizations of this semiclassical limit to a large class of covariant Schrödinger operators that act on sections in Hermitian vector bundle over $(X,m,b)$, a result that particularly applies to magnetic Schrödinger operators that are defined on $(X,m,b)$.

References in corpus (5)

• Convergence of time-inhomogeneous geodesic random walks and its application to coupling methods
• Self-adjoint extensions of discrete magnetic Schrödinger operators
• A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs
• Spectral band localization for Schrödinger operators on periodic graphs
• The Spectrum of Non-Local Discrete Schroedinger Operators with a delta-Potential

Cited by in corpus (3)

• A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs
• Heat kernel estimates and the relative compactness of perturbations by potentials
• Laplacians on periodic graphs with guides