Continuum limits for discrete Dirac operators on 2D square lattices

09/09/2021

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by   Karl Michael Schmidt, et al.

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We discuss the continuum limit of discrete Dirac operators on the square lattice in ℝ^2 as the mesh size tends to zero. To this end, we propose a natural and simple embedding of ℓ^2(ℤ_h^d) into L^2(ℝ^d) that enables us to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space L^2(ℝ^2)^2. In particular, we prove strong resolvent convergence. Potentials are assumed to be bounded and uniformly continuous functions on ℝ^2 and allowed to be complex matrix-valued.