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.
"On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I." Commun. Math. Sci. 4 (4) 741 - 766, December 2006.