Spurious localized highest-frequency modes in Schr\"odinger-type equations solved by finite-difference methods

TL;DR

This paper explains why finite-difference methods can produce spurious localized high-frequency modes in Schrödinger-type equations, which are numerical artifacts not present in the true solutions.

Contribution

The paper identifies and explains the origin of localized high-frequency modes as a numerical artifact in finite-difference solutions of Schrödinger-type equations.

Findings

Finite-difference methods can produce spurious localized high-frequency modes.

These localized modes are numerical artifacts, not physical solutions.

The paper provides an explanation for the emergence of these artifacts.

Abstract

High-frequency solutions of one or several Schr\"odinger-type equations are well known to differ very little from the plane wave solutions $\exp[\pm ik x]$. That is, the potential terms impact the envelope of a high-frequency plane wave by only a small amount. However, when such equations are solved by a finite-difference method, the highest-frequency solutions may, under certain conditions, turn out to be localized. In this letter we explain this numerical artifact.