The time-dependent Schrödinger equation of dimension k + 1: explicit and rational solutions via GBDT and multinodes

A version of the binary Darboux transformation is constructed for a non-stationary Schrodinger equation of dimension k + 1, where k > or = 1 is the number of space variables. This is an iterated generalized Backlund-Darboux transformation version. New families of non-singular and rational potentials and solutions are obtained. Some results are also new for the case that k = 1. A certain generalization of a colligation introduced by M S Livsic and a generalization of the S-node introduced by L A Sakhnovich have been successfully used in our construction.