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Continuum limit for lattice Schrödinger operators

We study the behavior of solutions of the Helmholtz equation $(- Δ_{disc,h} - E)u_h = f_h$ on a periodic lattice as the mesh size $h$ tends to 0. Projecting to the eigenspace of a characteristic root $λ_h(ξ)$ 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 $λ_h(ξ) \to P(ξ)$. 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{ö}dinger equation $( - Δ_{disc,h} +V_{disc,h} - E)u_h = f_h$ converges to that of the continuum Schr{ö}dinger equation $(P(D_x) + V(x) -E)u = f$.