Continuum limit for lattice Schr\"odinger operators

We study the behavior of solutions of the Helmholtz equation $(- \Delta_{disc,h} - E)u_h = f_h$onaperiodiclatticeasthemeshsize$h$ 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 ${\mathbf{R}}^d$, 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 $(-{\mathrm{\Delta }}_{disc,h}+V_{disc,h}-E)u_h=f_h$ converges to that of the continuum Schr{"o}dinger equation $(P(D_x)+V(x)-E)u=f$.