Discrete approximations to Dirac operators and norm resolvent convergence

TL;DR

This paper investigates the convergence of discrete Dirac operators to their continuous versions, showing that certain modifications ensure norm resolvent convergence across dimensions, which is crucial for accurate numerical approximations.

Contribution

It demonstrates that standard finite difference schemes do not always achieve norm resolvent convergence, but a simple modification involving the mass term guarantees this convergence.

Findings

Forward-backward differences yield norm resolvent convergence in 1D.

Symmetric differences do not lead to norm resolvent convergence in any dimension.

Adding a mass term to the discrete models ensures norm resolvent convergence generally.

Abstract

We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, H\"older continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not. The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.