Continuum Limits for Discrete Dirac Operators on 2D Square Lattices
Explore a mathematical lecture on the continuum limits of discrete Dirac operators on 2D square lattices. Delve into the proposed embedding of $\ell^2(\mathbb Z_h^d)$ into $L^2(\mathbb R^d)$, enabling the comparison of discrete and continuum Dirac operators in $L^2(\mathbb R^2)^2$. Examine the proof of strong resolvent convergence for discrete Dirac operators to continuum Dirac operators, considering bounded and uniformly continuous potentials on $\mathbb R^2$. Investigate the lack of norm resolvent convergence and its connection to the absence of the Liouville theorem in discrete complex analysis. This 41-minute talk, presented by Tomio Umeda at the Erwin Schrödinger International Institute for Mathematics and Physics, was part of the Workshop on "Spectral Theory of Differential Operators in Quantum Theory" in November 2022.