Continuum limit of the lattice quantum graph Hamiltonian

TL;DR

This paper demonstrates that the spectrum and eigenfunctions of the lattice quantum graph Hamiltonian converge to those of the continuum Schr"odinger operator as the lattice spacing approaches zero, bridging discrete and continuous quantum models.

Contribution

It establishes the continuum limit of the lattice quantum graph Hamiltonian using recent results on discrete Schr"odinger operators, providing a rigorous connection between discrete and continuous quantum systems.

Findings

Spectrum converges to the continuum Schr"odinger operator

Eigenfunctions and eigenprojections also converge

Uses recent results on discrete Schr"odinger operator continuum limit

Abstract

We consider the quantum graph Hamiltonian on the square lattice in Euclidean space, and we show that the spectrum of the Hamiltonian converges to the corresponding Schr\"odinger operator on the Euclidean space in the continuum limit, and that the corresponding eigenfunctions and eigenprojections also converge in some sense. We employ the discrete Schr\"odinger operator as the intermediate operator, and we use a recent result by the second and third author on the continuum limit of the discrete Schr\"odinger operator.