On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I

Abstract

We consider initial-boundary value problems for a generalized time-dependent Schrödinger equation in $1D$ on the semi-axis and in $2D$ on a semi-bounded strip. For Crank-Nicolson finite-difference schemes, we suggest an alternative coupling to approximate transparent boundary conditions and present a condition ensuring unconditional stability. In the case of discrete transparent boundary conditions, we revisit the statement and the proof of stability together with the derivation of the conditions.