On one semidiscrete Galerkin method for a generalized time-dependent 2D Schrödinger equation

An initial–boundary value problem for a generalized 2D Schrödinger equation in a rectangular domain is considered. Approximate solutions of the form c1(x1, t)χ1(x1, x2) + · · ·+ cN(x1, t)χN(x1, x2) are treated, where χ1, . . . ,χN are the first N eigenfunctions of a 1D eigenvalue problem in x2 depending parametrically on x1 and c1, . . . , cN are coefficients to be defined; they are of interest for nuclear physics problems. The corresponding semidiscrete Galerkin approximate problem is stated and analyzed. Uniform-in-time error bounds of arbitrarily high orders O ( N log N ) in L2 and O ( N−(θ−1) log N ) in H1, θ > 1, are proved. © 2008 Elsevier Ltd. All rights reserved.