Spectral Properties of Some Degenerate Elliptic Differential Operators

TL;DR

This paper extends classical spectral criteria to degenerate elliptic operators, analyzing how boundary coefficient degeneracy affects the essential spectrum and providing conditions for purely discrete spectra in various domains.

Contribution

It introduces new criteria for the essential spectrum of degenerate elliptic operators, incorporating boundary degeneracy and Hardy inequalities, with applications to diverse domain geometries.

Findings

Criteria for lower bounds of the essential spectrum in degenerate cases

Conditions under which the spectrum is purely discrete

Application of Hardy inequalities to spectral analysis

Abstract

In this paper we extend classical criteria for determining lower bounds for the least point of the essential spectrum of second-order elliptic differential operators on domains $\Omega\subset\R^n$ allowing for degeneracy of the coefficients on the boundary. We assume that we are given a sesquilinear form and investigate the degree of degeneracy of the coefficients near $\partial\Omega$ that can be tolerated and still maintain a closable sesquilinear form to which the First Representation Theorem can be applied. Then, we establish criteria characterizing the least point of the essential spectrum of the associated differential operator in these degenerate cases. Applications are given for convex and non-convex $\Omega$ using Hardy inequalities, which recently have been proven in terms of the distance to the boundary, showing the spectra to be purely discrete.