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

An initial–boundary value problem for a generalized 2D Schrödinger equation in a rectangular domain is considered. Approximate solutions of the form c 1 ( x 1 , t ) χ 1 ( x 1 , x 2 ) + ⋯ + c N ( x 1 , t ) χ N ( x 1 , x 2 ) are treated, where χ 1 , … , χ 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 O ( N − θ log N ) in L 2 and O ( N − ( θ − 1 ) log 1 / 2 N ) in H 1 , θ > 1 , are proved.