Continuum limit for lattice Schr\'odinger operators

We study the behavior of solutions of the Helmholtz equation $(- \Delta_{disc,h} - E)u_h = f_h$ on a periodic lattice as the mesh size $h$ tends to 0. Projecting to the eigenspace of a characteristic root $\lambda_h(\xi)$ and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution $u_h$ converges to that for the equation $(P(D_x) - E)v = g$ for a continuous model on ${\bf R}^d$, where $\lambda_h(\xi) \to P(\xi)$. 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 Schr{\"o}dinger equation $( - \Delta_{disc,h} +V_{disc,h} - E)u_h = f_h$ converges to that of the continuum Schr{\"o}dinger equation $(P(D_x) + V(x) -E)u = f$.