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[±ikx]exp[±ikx]. That is, the potential terms impact the envelope of a high-frequency plane wave to only a small extent. However, when such equations are solved by a finite-difference method, the highest-frequency solutions may, under certain conditions, turn out to be localized. This may puzzle the researcher and suggest that the code may have an error. However, this is not an error but a numerical artifact, and in this note we explain it.