Continuum limit for lattice Schrödinger operators

We study the behavior of solutions of the Helmholtz equation (-Δdisc,h - E)uh = fh on a periodic lattice as the mesh size h tends to 0. Projecting to the eigenspace of a characteristic root λh(ζ) and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution uh converges to that for the equation (P(Dx) - E)v = g for a continuous model on Rd, where λh(ζ) → P(ζ). For the case of the hexagonal and related lattices, in a suitable energy region, it converges to that for the Dirac equation. For the case of the square lattice, triangular lattice, hexagonal lattice (in another energy region) and subdivision of a square lattice, one can add a scalar potential, and the solution of the lattice Schrödinger equation (-Δdisc,h + Vdisc,h - E)uh = fh converges to that of the continuum Schrödinger equation (P(Dx) + V (x) - E)u = f.