On the resolvent of the Dirac operator in ${\mathbb{R}}^2$

Abstract

In the present paper, we prove an abstract functional analytic criterion for a class of linear partial differential operators acting on a domain $\Omega \subseteq {\mathbb{R}}^n$ which are elliptic in the interior to have compact resolvent. This extends known results for magnetic Schrödinger 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 $\mathbb{C}$ and we use this connection to prove a non-compactness result for its resolvent.

References in corpus (1)

• Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces

Cited by in corpus (2)

• An idea on proving weighted Sobolev embeddings
• Compactness of the d-bar-Neumann operator on singular complex spaces