ABC of ladder operators for rationally extended quantum harmonic oscillator systems

Abstract

The problem of construction of ladder operators for rationally extended quantum harmonic oscillator (REQHO) systems of a general form is investigated in the light of existence of different schemes of the Darboux-Crum-Krein-Adler transformations by which such systems can be generated from the quantum harmonic oscillator. Any REQHO system is characterized by the number of separated states in its spectrum, the number of `valence bands' in which the separated states are organized, and by the total number of the missing energy levels and their position. All these peculiarities of a REQHO system are shown to be detected and reflected by a trinity $({\mathcal{A}}^{\pm }$, ${\mathcal{B}}^{\pm }$, ${\mathcal{C}}^{\pm }$) of the basic (primary) lowering and raising ladder operators related between themselves by certain algebraic identities with coefficients polynomially-dependent on the Hamiltonian. We show that all the secondary, higher-order ladder operators are obtainable by a composition of the basic ladder operators of the trinity which form the set of the spectrum-generating operators. Each trinity, in turn, can be constructed from the intertwining operators of the two complementary minimal schemes of the Darboux-Crum-Krein-Adler transformations.

29 pages, 1 figure; comments and refs added, version to appear in J Phys A

References in corpus (13)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Self-isospectrality, special supersymmetry, and their effect on the band structure
• Solvable rational extensions of the isotonic oscillator
• Aharonov-Bohm effect on AdS_2 and nonlinear supersymmetry of reflectionless Poschl-Teller system
• Group theoretical approach to the intertwined Hamiltonians
• Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems
• The generalized quantum isotonic oscillator
• Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator
• Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
• Exactly Solvable Quantum Mechanics

Cited by in corpus (16)

• Conformal bridge between asymptotic freedom and confinement
• Rational deformations of conformal mechanics
• Hidden symmetries of rationally deformed superconformal mechanics
• Klein four-group and Darboux duality in conformal mechanics
• Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group
• Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background
• Nonlinear symmetries of perfectly invisible $PT$-regularized conformal and superconformal mechanics systems
• Generalized ladder operators for the perturbed harmonic oscillator
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator
• Nonlinear supersymmetry as a hidden symmetry
• Harmonic Oscillator with a Step and its Isospectral Properties
• Hallmarks of tunneling dynamics with broken reflective symmetry
• Exotic nonlinear supersymmetry and integrable systems
• Recurrence relations and general solution of the exceptional Hermite equation
• Hidden symmetries and nonlinear (super)algebras