I would like to 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, I propose the most natural and simplest embedding of ell^2( Z^d_h ) into L^2( R^d).
This approach enables one to define difference operators in a subspace of step functions in L^2( R^d) and to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space L^2(R^2)^2. In particular, I can prove that 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 R^2 and allowed to be complex matrix-valued. I also prove that the discrete Dirac operators do not converge to the continuum Dirac operators in the norm resolvent sense.
This finding is closely related to the observation that the Liouville theorem does not hold in discrete complex analysis. If time allows, I would also like to discuss the continuum limit of discrete Dirac operators on the 1D lattice.
(This is joint work with Karl Michael Schmidt.)
