Resolvent convergence to Dirac operators on planar domains

TL;DR

This paper proves that a Dirac operator with a large mass term outside a domain converges in norm resolvent sense to a Dirac operator with infinite mass boundary conditions inside the domain, applicable to various domain types.

Contribution

It provides a simple proof of resolvent convergence for Dirac operators with mass terms, extending to unbounded domains and including external potentials.

Findings

Norm resolvent convergence established as mass parameter tends to infinity.

Convergence results hold for both bounded and unbounded domains.

Method extends to operators with external matrix-valued potentials.

Abstract

Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Omega. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Omega with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.