N-order Darboux transformation and a spectral problem on semiaxis

TL;DR

This paper introduces an N-order Darboux transformation operator, explores its properties, and applies it to construct potentials with bound states at arbitrary energies for a spectral problem on the semiaxis.

Contribution

It develops a general framework for N-order Darboux transformations and applies it to spectral problems on the semiaxis, enabling the construction of specific potentials.

Findings

Factorisation properties of the N-order Darboux operator are established.

Potentials with bound states at arbitrary energies are constructed.

The technique enhances the analysis of spectral problems on the semiaxis.

Abstract

N-order Darboux transformation operator is defined on the basis of a general notion of transformation operators. Factorisation properties of this operator are studied. The Darboux transformation operator technique is applied to construct and investigate potentials with bound states at arbitrary energies for the spectral problem on semiaxis.