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

An initial-boundary value problem for a generalized 2D Schrodinger equation in a rectangular domain is considered. Approximate solutions of the form c(1)(x(1),t) chi(1) (x(1),x(2)) + ... + c(N)(x(1),t) chi(N)(x(1),x(2)) are treated, where chi(1),..., chi(N) are the first N eigenfunctions of a 1D eigenvalue problem in X-2 depending parametrically on x(1) and c(1), ..., c(N) 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 0 (N-theta log N) in L-2 and 0 (N-((theta-1)) log(1/2) N) in H-1, theta > 1, are proved. (C) 2008 Elsevier Ltd. All rights reserved.