Perturbation of essential spectra of exterior elliptic problems

TL;DR

This paper investigates how different boundary conditions affect the essential spectrum of elliptic operators on exterior domains, showing that certain non-elliptic conditions can enlarge the spectrum, with extensions to higher-order operators and spectral asymptotics.

Contribution

It demonstrates that non-elliptic Neumann-type boundary conditions can augment the essential spectrum of elliptic operators, extending spectral asymptotics formulas and analysis techniques.

Findings

Non-elliptic boundary conditions can enlarge the essential spectrum.

Spectral asymptotics formulas are extended to higher-order operators.

Krein-type formulas and cutoff techniques are used in proofs.

Abstract

For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can augment, but not diminish, the essential spectrum by imposition of other Neumann-type non-elliptic boundary conditions. - The results are extended to 2m-order operators, where it is shown that for any selfadjoint realization defined by an elliptic normal boundary condition (other than the Dirichlet condition), one can augment the essential spectrum at will by adding a suitable operator to the mapping from free Diriclet data to Neumann data. We here also show an extension of the spectral asymptotics formula for the difference between inverses of elliptic problems. - The proofs rely on Krein-type formulas for differences between inverses, and cutoff techniques, combined with results on singular Green operators and their spectral asymptotics.