Extension theory for elliptic partial differential operators with pseudodifferential methods

TL;DR

This paper surveys how extension theories combined with pseudodifferential methods enhance understanding of elliptic operators on smooth domains, focusing on boundary problems, spectral asymptotics, and operator realizations.

Contribution

It demonstrates the application of pseudodifferential boundary problem theory to elliptic operator extensions, including recent results on spectral and boundedness properties.

Findings

Application of pseudodifferential boundary problem theory to elliptic operators

Results on lower boundedness of operator realizations

Spectral asymptotics for resolvent differences

Abstract

This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R^n, n > 1. The theory of pseudodifferential boundary problems has turned out to be very useful here, not only as a formulational framework, but also for the solution of specific questions. We recall some elements of that theory, and show its application in several cases (including recent results), namely to the lower boundedness question, and the question of spectral asymptotics for differences between resolvents.