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)c1(x1,t)χ1(x1,x2)+⋯+cN(x1,t)χN(x1,x2) are treated, where χ1,…,χNχ1,…,χN are the first NN eigenfunctions of a 1D eigenvalue problem in x2x2 depending parametrically on x1x1 and c1,…,cNc1,…,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−θlogN)O(N−θlogN) in L2L2 and O(N−(θ−1)log1/2N)O(N−(θ−1)log1/2N) in H1H1, θ>1θ>1, are proved.