On the resolvent of the Dirac operator in $\Bbb R^2$

TL;DR

This paper establishes conditions under which certain elliptic differential operators, including the Dirac operator in two dimensions, have compact resolvent, extending previous results for magnetic Schrödinger operators and exploring their connections to complex analysis.

Contribution

It provides an abstract criterion for compactness of resolvents for a broad class of elliptic operators, and relates the Dirac operator in 2D to the ar-Laplacian, demonstrating non-compactness in this context.

Findings

Extended known results to more general differential operators.

Established a connection between the Dirac operator and the ar-Laplacian.

Proved a non-compactness result for the resolvent of the Dirac operator.

Abstract

In the present paper, we prove an abstract functional analytic criterion for a class of linear partial differential operators acting on a domain $\Omega\subseteq\Bbb R^n$ which are elliptic in the interior to have compact resolvent. This extends known results for magnetic Schr\"{o}dinger operators to more general differential operators. We point out the relationship between the Dirac operator in real dimension two and the $\bar\partial$-Laplacian on a certain weighted space on $\Bbb C$ and we use this connection to prove a non-compactness result for its resolvent.