Group theoretical approach to the intertwined Hamiltonians

Abstract

We show that the finite difference Bäcklund formula for the Schrödinger Hamiltonians is a particular element of the transformation group on the set of Riccati equations considered by two of us in a previous paper. Then, we give a group theoretical explanation to the problem of Hamiltonians related by a first order differential operator. A generalization of the finite difference algorithm relating eigenfunctions of {\emph three} different Hamiltonians is found, and some illustrative examples of the theory are analyzed, finding new potentials for which one eigenfunction and its corresponding eigenvalue is exactly known.

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• A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour
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• Intertwining technique for the one-dimensional stationary Dirac equation
• Coherent states for Hamiltonians generated by supersymmetry
• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• Extended Krein-Adler theorem for the translationally shape invariant potentials
• Second order SUSY transformations with `complex energies'
• Intertwined Hamiltonians in Two Dimensional Curved Spaces
• Multistep DBT and regular rational extensions of the isotonic oscillator
• Solvable rational extensions of the Morse and Kepler-Coulomb potentials
• The Phenomenon of Darboux Displacements
• Wronskian formula for confluent second-order supersymmetric quantum mechanics
• New rational extensions of solvable potentials with finite bound state spectrum
• Supersymmetric partners of the trigonometric Poschl-Teller potentials
• Trends in supersymmetric quantum mechanics
• The Quantum Mechanics SUSY Algebra: An Introductory Review
• Non-Hermitian SUSY Hydrogen-like Hamiltonians with real spectra
• Rational solutions for the Riccati-Schrödinger equations associated to translationally shape invariant potentials
• Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator
• Confluent Chains of DBT: Enlarged Shape Invariance and New Orthogonal Polynomials
• ABC of ladder operators for rationally extended quantum harmonic oscillator systems
• On the new translational shape invariant potentials
• Supersymmetric partners of the harmonic oscillator with an infinite potential barrier
• SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations
• Coherent states for the supersymmetric partners of the truncated oscillator
• Symmetries and the compatibility condition for the new translational shape invariant potentials
• A geometric approach to time evolution operators of Lie quantum systems
• Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator
• Higher Order Supersymmetric Truncated Oscillators
• An affine Weyl group characterization of polynomial Heisenberg algebras
• Solution of Second Order Supersymmetrical Intertwining Relations in Minkowski Plane
• Stratified Lie systems: Theory and applications