Spurious localized highest-frequency modes in Schrödinger-type equations solved by finite-difference methods

High-frequency solutions of one or several Schrödinger-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.

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