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.

翻译：摘要：本文讨论了当网格尺寸趋于零时，$\mathbb R^2$ 中正方晶格上离散狄拉克算子的连续极限。为此，我们提出了最自然且最简单的将 $\ell^2(\mathbb Z_h^d)$ 嵌入 $L^2(\mathbb R^d)$ 的方法，从而能够在同一希尔伯特空间 $L^2(\mathbb R^2)^2$ 中比较离散狄拉克算子与连续狄拉克算子。特别地，我们证明离散狄拉克算子在强预解意义下收敛到连续狄拉克算子。势函数假定为 $\mathbb R^2$ 上的有界且一致连续函数，并允许取复矩阵值。我们还证明离散狄拉克算子不会在范数预解意义下收敛到连续狄拉克算子，这一结论与离散复分析中刘维尔定理不成立的现象密切相关。