Continuum limits for discrete Dirac operators on 2D square lattices

TL;DR

This paper investigates how discrete Dirac operators on 2D square lattices approach their continuum counterparts as the lattice mesh size shrinks, establishing strong resolvent convergence but not norm resolvent convergence.

Contribution

It introduces a natural embedding of lattice functions into continuous space and proves strong resolvent convergence of discrete to continuum Dirac operators.

Findings

Discrete Dirac operators converge strongly to continuum operators.

Convergence in norm resolvent sense does not hold.

The approach clarifies the relationship between discrete and continuous Dirac operators.

Abstract

We discuss the continuum limit of discrete Dirac operators on the square lattice in $\mathbb R^2$ as the mesh size tends to zero. To this end, we propose the most natural and simplest embedding of $\ell^2(\mathbb Z_h^d)$ into $L^2(\mathbb R^d)$, which enables us to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space $L^2(\mathbb R^2)^2$. In particular, we prove that the discrete Dirac operators converge to the continuum Dirac operators in the strong resolvent sense. Potentials are assumed to be bounded and uniformly continuous functions on $\mathbb R^2$ and allowed to be complex matrix-valued. We also prove that the discrete Dirac operators do not converge to the continuum Dirac operators in the norm resolvent sense. This is closely related to the observation that the Liouville theorem does not hold in discrete complex analysis.