Continuum limit for a discrete Hodge-Dirac operator on square lattices

Abstract

We study the continuum limit for Dirac-Hodge operators defined on the $n$ dimensional square lattice $h{\mathbb{Z}}^n$ as $h$ goes to $0$. This result extends to a first order discrete differential operator the known convergence of discrete Schrödinger operators to their continuous counterpart. To be able to define such a discrete analog, we start by defining an alternative framework for a higher-dimensional discrete differential calculus. We believe that this framework, that generalize the standard one defined on simplicial complexes, could be of independent interest. We then express our operator as a differential operator acting on discrete forms to finally be able to show the limit to the continuous Dirac-Hodge operator.

References in corpus (7)

• Location of eigenvalues of non-self-adjoint discrete Dirac operators
• One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics
• Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals
• Continuum limit of the lattice quantum graph Hamiltonian
• Norm resolvent convergence of discretized Fourier multipliers
• Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence
• Spectral Gap of The Discrete Laplacian On Triangulations

Cited by in corpus (3)

• Complex Quantum Networks: a Topical Review
• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• Continuum limit for Laplace and Elliptic operators on lattices