Continuum limit for lattice Schrödinger operators

Hiroshi Isozaki, Arne Jensen

Research output: Working paperResearch

Abstract

We study the behavior of solutions of the Helmholtz equation (−Δdisc,h−E)uh=fh on a periodic lattice as the mesh size h tends to 0. Projecting to the eigenspace of a characteristic root λh(ξ) and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution uh converges to that for the equation (P(Dx)−E)v=g for a continuous model on Rd, where λh(ξ)→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 (−Δdisc,h+Vdisc,h−E)uh=fh converges to that of the continuum Schr{ö}dinger equation (P(Dx)+V(x)−E)u=f.
Original languageEnglish
PublisherarXiv.org
Publication statusPublished - 1 Jun 2020

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