TY - UNPB
T1 - Continuum limit for lattice Schrödinger operators
AU - Isozaki, Hiroshi
AU - Jensen, Arne
PY - 2020/6/1
Y1 - 2020/6/1
N2 - 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.
AB - 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.
UR - https://arxiv.org/abs/2006.00854
M3 - Working paper
BT - Continuum limit for lattice Schrödinger operators
PB - arXiv
ER -