TY - JOUR
T1 - Continuum limit for lattice Schrödinger operators
AU - Isozaki, Hiroshi
AU - Jensen, Arne
N1 - Publisher Copyright:
© 2021 World Scientific Publishing Company.
PY - 2022
Y1 - 2022
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(ζ) 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(ζ) → P(ζ). 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 Schrödinger equation (-Δdisc,h + Vdisc,h - E)uh = fh converges to that of the continuum Schrödinger 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(ζ) 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(ζ) → P(ζ). 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 Schrödinger equation (-Δdisc,h + Vdisc,h - E)uh = fh converges to that of the continuum Schrödinger equation (P(Dx) + V (x) - E)u = f.
KW - lattice
KW - scattering theory
KW - Schrödinger operator
UR - http://www.scopus.com/inward/record.url?scp=85117108001&partnerID=8YFLogxK
U2 - 10.1142/S0129055X22500015
DO - 10.1142/S0129055X22500015
M3 - Journal article
AN - SCOPUS:85117108001
SN - 0129-055X
VL - 34
JO - Reviews in Mathematical Physics
JF - Reviews in Mathematical Physics
IS - 2
M1 - 2250001
ER -