# Continuum limits for lattice Schrödinger equations

Hiroshi Isozaki
University of Tsukuba
https://www.researchgate.net/profile/Hiroshi-Isozaki

Date(s) : 14/02/2023   iCal
11 h 00 min - 12 h 00 min

We consider the behavior of solutions of the Helmholtz equation

\$(- \Delta_{disc,h} – E)u_h = f_h\$ for a continuous spectrum \$E\$ on a periodic lattice as the mesh size \$h\$ tends to 0. 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 Schrödinger equation \$(P(D_x) + V(x) -E)u = f\$. This is a joint work with A. Jensen.

https://arxiv.org/abs/2006.00854

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