New exactly solvable reflectionless potentials of Gamov's type

Abstract

In paper SUSY-hierarchies of one-dimensional potentials with continuous energy spectra are studied. Use of such hierarchies for analysis of reflectionless potentials is substantiated from the physical point of view. An interdependence (based on Darboux transformations) between spectral characteristics of potentials-partners is determined, an uniqueness of its solution in result of use of boundary conditions is shown. A rule of construction of new reflectionless potentials on the basis of known one is corrected, its proof is proposed. At first time a general solution for a superpotential $W_{n+m}(x)$ with an arbitrary number n+m in the studied hierarchy on the basis of only one known partial solution for the superpotential $W_{n}(x)$ with the selected number n is found. A general solution of a hierarchy of inverse power (reflectionless) potentials is obtained. Such a hierarchy can be interesting as an example for solution of a known problem of search of general solutions of the hierarchies of different types (both in standard and parasupersymmetric quantum mechanics). A consequent statement and analysis of exactly solvable reflectionless potentials of Gamov's type, which at their shapes look qualitatively like scattering potentials in two-particle description of collisions between particles and nuclei or decay potentials in two-particle description of decay of compound spherical nuclear systems, are presented.