Second Order Darboux Displacements

Abstract

The potentials for a one dimensional Schroedinger 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 Schroedinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential. It is proven that a particular case of the periodic Lame-Ince potential is 2-SUSY invariant. Both Bloch solutions of the corresponding Schroedinger equation equation are found for any value of the energy. A simple analytic expression for a family of two-gap potentials is derived.

References in corpus (4)

• Self-Isospectral Periodic Potentials and Supersymmetric Quantum Mechanics
• New Solvable and Quasi Exactly Solvable Periodic Potentials
• Nonlocal supersymmetric deformations of periodic potentials
• The Phenomenon of Darboux Displacements

Cited by in corpus (7)

• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• Finite-gap systems, tri-supersymmetry and self-isospectrality
• Periodic Potentials and Supersymmetry
• Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity
• A Class of Partially Solvable Two-Dimensional Quantum Models with Periodic Potentials
• Absence of Floquet scattering in oscillating non-Hermitian potential wells
• Supersymmetric partners for the associated Lame potentials