Continuum limit for lattice Schrödinger operators

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 ${\mathbf{R}}^d$, 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{ö}dinger equation $(-{\Delta }_{disc,h}+V_{disc,h}-E)u_h=f_h$ converges to that of the continuum Schr{ö}dinger equation $(P(D_x)+V(x)-E)u=f$.

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• Spectral Properties of Schrödinger Operators on Perturbed Lattices
• Inverse scattering for Schrödinger operators on perturbed lattices
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• Branching form of the resolvent at threshold for multi-dimensional discrete Laplacians
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Cited by in corpus (4)

• Continuum limit of the lattice quantum graph Hamiltonian
• Norm resolvent convergence of discretized Fourier multipliers
• Continuum limits for discrete Dirac operators on 2D square lattices
• Semiclassical analysis and the Agmon-Finsler metric for discrete Schrödinger operators