SUSY-hierarchy of one-dimensional reflectionless potentials

Abstract

A class of one-dimensional reflectionless potentials is studied. It is found that all possible types of the reflectionless potentials can be combined into one SUSY-hierarchy with a constant potential. 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 integral form of interdependence between superpotentials with neighboring numbers of this hierarchy, opening a possibility to find new reflectionless potentials, is found and has a simple analytical view. It is supposed that any possible type of the reflectionless potential can be expressed through finite number of elementary functions (unlike some presentations of the 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). 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)=±α/|x−x0|n (where α and x0 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 are analyzed.