Continuum limit for Laplace and Elliptic operators on lattices

Abstract

Continuum limits of Laplace operators on general lattices are considered, and it is shown that these operators converge to elliptic operators on the Euclidean space in the sense of the generalized norm resolvent convergence. We then study operators on the hexagonal lattice, which does not apply the above general theory, but we can show its Laplace operator converges to the continuous Laplace operator in the continuum limit. We also study discrete operators on the square lattice corresponding to second order strictly elliptic operators with variable coefficients, and prove the generalized norm resolvent convergence in the continuum limit.

References in corpus (5)

• Continuum limit of the lattice quantum graph Hamiltonian
• Continuum limit for a discrete Hodge-Dirac operator on square lattices
• Long-range scattering theory for discrete Schrödinger operators on graphene
• Continuum limits for discrete Dirac operators on 2D square lattices
• Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence