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Second-order Darboux displacement. (English) Zbl 1049.81019

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
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