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.
@article{1175797609,
author = {Ducomet, B. and Zlotnik, A.},
title = {On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schr\"odinger equation. I},
journal = {Commun. Math. Sci.},
volume = {4},
number = {1},
year = {2006},
pages = { 741-766},
language = {en},
url = {http://dml.mathdoc.fr/item/1175797609}
}
Ducomet, B.; Zlotnik, A. On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I. Commun. Math. Sci., Tome 4 (2006) no. 1, pp. 741-766. http://gdmltest.u-ga.fr/item/1175797609/