SUSY-hierarchy of one-dimensional reflectionless potentials

TL;DR

This paper investigates a class of one-dimensional reflectionless potentials within a SUSY-hierarchy framework, providing a method to construct new potentials with simple analytical forms and analyzing their transparency properties, especially for inverse power potentials.

Contribution

It introduces a recurrent method to determine the general form of reflectionless potentials within a SUSY-hierarchy, enabling the construction of new potentials with explicit analytical expressions.

Findings

Reflectionless potentials can be expressed through elementary functions.

Inverse power potentials are reflectionless only for n=2.

A SUSY-hierarchy of inverse power reflectionless potentials is constructed.

Abstract

A class of one-dimensional reflectionless potentials, an absolute transparency of which is concerned with their belonging to one SUSY-hierarchy with a constant potential, is studied. An approach for determination of a general form of the reflectionless potential on the basis of construction of such a hierarchy by the recurrent method is proposed. A general form of interdependence between superpotentials with neighboring numbers of this hierarchy, opening a possibility to find new reflectionless potentials, have a simple analytical view and are expressed through finite number of elementary functions (unlike some reflectionless potentials, which are constructed on the basis of soliton solutions or are shape invariant in one or many steps with involving scaling of parameters, and are expressed through series), is obtained. An analysis of absolute transparency existence for the potential which has the inverse power dependence on space coordinate (and here tunneling is possible), i.e. which has the form $V(x) = \pm \alpha / |x-x_{0}|^{n}$ (where $\alpha$ and $x_{0}$ are constants, $n$ is natural number), is fulfilled. It is shown that such a potential can be reflectionless at n = 2 only. A SUSY-hierarchy of the inverse power reflectionless potentials is constructed. Isospectral expansions of this hierarchy is analyzed.