On an Inclusion of the Essential Spectrum of Laplacians under Non-Compact Change of Metric

TL;DR

This paper investigates how the essential spectrum of Laplacians on a Riemannian manifold is affected by non-compact changes in the metric, including cases with singular or fractal-like subsets.

Contribution

It establishes the stability of essential self-adjointness and describes the inclusion relations of essential spectra under non-compact metric modifications.

Findings

Essential self-adjointness remains stable under certain non-compact metric changes.

The essential spectrum of Laplacians is included under metric modifications on subsets with possibly infinite volume.

The results apply even when the metric change involves singular or fractal sets.

Abstract

It is shown the stability of the essential self-adjointness, and an inclusion of the essential spectra of Laplacians under the change of Riemannian metric on a subset K of M. The set K may have infinite volume measured with the new metric and its completion may contain a singular set such as fractal, to which the metric is not extendable.