Comment on "Bound states of edge dislocations: The quantum dipole problem in two dimensions"

TL;DR

This paper critically re-evaluates previous numerical results on quantum bound states near edge dislocations, using a refined Rayleigh-Ritz method to provide more accurate energy bounds and state calculations.

Contribution

It offers a rigorous upper bound for the ground state energy and precise excited state energies, correcting and improving upon prior numerical approaches.

Findings

The ground state energy bound is more accurate than previous estimates.

Excited state energies up to the 500th level match expected asymptotic behavior.

The Rayleigh-Ritz method provides a reliable framework for this quantum problem.

Abstract

We show that the numerical results contained in a recent paper are affected by a non optimal implementation of the methods which have been used to obtain these results. A careful analysis done using the Rayleigh-Ritz method provides a rigorous upper bound for the energy of the ground state of an electron in a two dimensional potential generated by the edge dislocation, as well as precise values for the excited states. The extrapolation of the results corresponding to different subspaces is used to obtain a precise estimate of the fundamental energy of the model. The energies of the first 500 states that we have calculated are in perfect agreement with the expected asymptotic behavior.