Continuum limit for lattice Schr\"odinger operators

TL;DR

This paper investigates the continuum limit of solutions to lattice Schrödinger and Helmholtz equations, showing convergence to continuum models including Dirac and Schrödinger equations under various lattice structures.

Contribution

It establishes the convergence of lattice solutions to continuum equations, including Dirac and Schrödinger equations, for different lattice geometries and energy regions.

Findings

Lattice solutions converge to continuum models as mesh size tends to zero.

In hexagonal lattices, solutions converge to Dirac equations in certain energy regions.

For square and triangular lattices, solutions converge to Schrödinger equations with scalar potentials.

Abstract

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 $(P(D_x) + V(x) -E)u = f$.