Samsonov, B. F.; Glasser, M. L.; Negro, J.; Nieto, L. M. Second-order Darboux displacement. (English) Zbl 1049.81019 J. Phys. A, Math. Gen. 36, No. 39, 10053-10069 (2003). Summary: The potentials for a one-dimensional Schrödinger equation that are displaced along the \(x\)-axis under second-order Darboux transformations, called 2-SUSY invariant, are characterized in terms of a differential-difference equation. The solutions of the Schrödinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential. It is proved that a particular case of the periodic Lamé-Ince potential is 2-SUSY invariant. Both Bloch solutions of the corresponding Schrödinger equation are found for any value of the energy. A simple analytic expression for a family of two-gap potentials is derived. MSC: 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 82D20 Statistical mechanics of solids 81Q60 Supersymmetry and quantum mechanics 39A10 Additive difference equations PDFBibTeX XMLCite \textit{B. F. Samsonov} et al., J. Phys. A, Math. Gen. 36, No. 39, 10053--10069 (2003; Zbl 1049.81019) Full Text: DOI arXiv