Volume growth and bounds for the essential spectrum for Dirichlet forms

TL;DR

This paper investigates how the volume growth of spaces associated with Dirichlet forms influences the spectral bounds of related operators, including graph Laplacians, revealing thresholds for spectral positivity based on growth rates.

Contribution

It establishes bounds for the essential spectrum's bottom using intrinsic metric volume growth and identifies growth thresholds affecting spectral properties of graph Laplacians.

Findings

Bounds for the bottom of the spectrum in terms of exponential volume growth.

Threshold for positivity of the spectrum at cubic polynomial growth.

Finiteness of the essential spectrum's bottom linked to volume growth rates.

Abstract

We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric) we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth.