Time-dependent Schroedinger equation in dimension $k+1$: explicit and rational solutions via GBDT and multinodes

TL;DR

This paper develops an iterated GBDT method to construct explicit and rational solutions for the time-dependent Schrödinger equation in multiple dimensions, introducing new potentials and solutions, including for the one-dimensional case.

Contribution

It introduces a novel iterated GBDT approach for the multidimensional Schrödinger equation, generalizing colligation and S-node concepts to generate new solutions and potentials.

Findings

Constructed a binary Darboux transformation for the non-stationary Schrödinger equation.

Derived new families of non-singular and rational potentials and solutions.

Extended the approach to the one-dimensional case, providing new results.

Abstract

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