Supersymmetry, potentials with bound states at arbitrary energies and multi-soliton configurations

Abstract

The connection between the algebra of supersymmetry and the inverse scattering method is used to construct one-dimensional potentials with any specified number of non-degenerate bound states at arbitrary energies. The reflection coefficient of the potential so constructed is related to the reflection coefficient of a reference potential which supports no bound states. It is shown that, by choosing the reference potential to be V=0, it is possible to construct reflectionless potentials with bound states at arbitrary energies. The relationship of this construction based on supersymmetry to other known constructions of reflectionless potentials is established. It is shown that the symmetric reflectionless potential may be expressed as a linear combination of the squares of bound state eigenfunctions with coefficients related to the wavenumbers associated with the bound states.