Riccati equation, Factorization Method and Shape Invariance

Abstract

The basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed. The relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the solutions of Infeld and Hull are greatly generalized in a simple way.

Latex file, 35 pages

References in corpus (4)

• The nonlinear superposition principle and the Wei-Norman method
• Integrability of Riccati equation from a group theoretical viewpoint
• The factorization of the hypergeometric equation
• Shape Invariant potentials depending on n parameters transformed by translation

Cited by in corpus (40)

• Dirac oscillator with nonzero minimal uncertainty in position
• A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour
• One-dimensional model of a quantum nonlinear harmonic oscillator
• Superposition rules, Lie theorem and partial differential equations
• Group theoretical approach to the intertwined Hamiltonians
• Supersymmetry and algebraic deformations
• More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum
• Second order SUSY transformations with `complex energies'
• Dirac electron in graphene with magnetic fields arising from first-order intertwining operators
• Shape Invariant potentials depending on n parameters transformed by translation
• Recent Applications of the Theory of Lie Systems in Ermakov Systems
• 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
• Electron in bilayer graphene with magnetic fields leading to shape invariant potentials
• Inter-relations between additive shape invariant superpotentials
• ABC of ladder operators for rationally extended quantum harmonic oscillator systems
• On the new translational shape invariant potentials
• Supersymmetry identifies molecular Stark states whose eigenproperties can be obtained analytically
• Two important examples of nonlinear oscillators
• Method for Generating Additive Shape Invariant Potentials from an Euler Equation
• Exactness of SWKB for Shape Invariant Potentials
• A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
• Nonlinear superpositions and Ermakov systems
• Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method
• Connection between type B (or C) and F factorizations and construction of algebras
• A geometric approach to integrability conditions for Riccati equations
• A geometric approach to time evolution operators of Lie quantum systems
• Integrability of Lie systems through Riccati equations
• Symmetries and the compatibility condition for the new translational shape invariant potentials
• Shape invariance through Crum transformation
• Exactness of Semiclassical Quantization Rule for Broken Supersymmetry
• Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians
• On self-gravitating kinks in two-dimensional pseudo-riemannian universes
• Nonstationary deformed singular oscillator: quantum invariants and the factorization method
• Quantum-based solution of time-dependent complex Riccati equations
• Generalized Langer Correction and the Exactness of WKB for all Conventional Potentials
• A Venn diagram for supersymmetric, exactly solvable, shape invariant, and Infeld-Hull factorizable potentials
• Quantum Hamilton-Jacobi Quantization and Shape Invariance
• Infinite families of shape invariant potentials with n parameters subject to translation
• Applications of Lie systems in Quantum Mechanics and Control Theory