Higher-order supersymmetric quantum mechanics

Abstract

We review the higher-order supersymmetric quantum mechanics (H-SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first-order SUSY transformations are used to derive in a simple way the higher-order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H-SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.

42 pages, 12 eps figures

Cited by in corpus (69)

• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry
• Coherent states for Hamiltonians generated by supersymmetry
• Position Dependent Mass Oscillators and Coherent States
• Dirac electron in graphene under supersymmetry generated magnetic fields
• Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
• Novel Enlarged Shape Invariance Property and Exactly Solvable Rational Extensions of the Rosen-Morse II and Eckart Potentials
• Non-Hermitian Quantum Mechanics of Non-diagonalizable Hamiltonians: puzzles with self-orthogonal states
• New supersymmetry-generated complex potentials with real spectra
• New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials
• New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
• SUSY transformations between digonalizable and non-diagonalizable Hamiltonians
• Dirac electron in graphene with magnetic fields arising from first-order intertwining operators
• Factorization method and new potentials from the inverted oscillator
• Supersymmetric partners of the trigonometric Poschl-Teller potentials
• Trends in supersymmetric quantum mechanics
• Wronskian differential formula for confluent supersymmetric quantum mechanics
• Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability
• Jacobi photonic lattices and their SUSY partners
• Connection between quantum systems involving the fourth Painleve transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite EOP
• Position Dependent Mass Schroedinger Equation and Isospectral Potentials : Intertwining Operator approach
• Photonic systems with two-dimensional landscapes of complex refractive index via time-dependent supersymmetry
• Supersymmetric Quantum Mechanics and Painlevé IV Equation
• Electron in bilayer graphene with magnetic fields leading to shape invariant potentials
• Exactly solvable associated Lame potentials and supersymmetric transformations
• Bilayer graphene in magnetic fields generated by supersymmetry
• New supersymmetric partners for the associated Lame potentials
• Hyperconfluent third-order supersymmetric quantum mechanics
• The generalized zero-mode supersymmetry scheme and the confluent algorithm
• Solutions of the Dirac Equation in a Magnetic Field and Intertwining Operators
• Supersymmetric partners of the harmonic oscillator with an infinite potential barrier
• Irreducible second order SUSY transformations between real and complex potentials
• Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum
• Solutions to the Painlevé V equation through supersymmetric quantum mechanics
• Complex oscillator and Painlevé IV equation
• SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations
• Confluent Second-Order Supersymmetric Quantum Mechanics and Spectral Design
• SU(1,1) and SU(2) Approaches to the Radial Oscillator: Generalized Coherent States and Squeezing of Variances
• Exact propagators for SUSY partners
• Coherent states for the supersymmetric partners of the truncated oscillator
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• Nonsingular potentials from excited state factorization of a quantum system with position dependent mass
• Complex Supersymmetry in Graphene
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Pöschl-Teller Hamiltonian: Gazeau-Klauder type coherent states, related statistics and geometry
• Coherent states for quadratic Hamiltonians
• Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
• Extended WKB method, resonances and supersymmetric radial barriers
• CPT-conserved effective mass Hamiltonians through first and higher order charge operator C in a supersymmetric framework
• Strongly interacting matter in extreme magnetic fields
• Supersymmetry Transformations for Delta Potentials
• Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
• Harmonic Oscillator SUSY Partners and Evolution Loops
• Exact Propagators for Soliton Potentials
• Instanton-Induced Supersymmetry Breaking in Topological Semimetals
• Supersymmetric Quantum Potentials Analogs of Classical Electrostatic Fields
• Darboux transformations and second order difference equations
• Supersymmetric Quantum Mechanics, multiphoton algebras and coherent states
• An affine Weyl group characterization of polynomial Heisenberg algebras
• Disconjugacy, regularity of multi-indexed rationally-extended potentials, and Laguerre exceptional polynomials
• Freezable bound states in the continuum for time-dependent quantum potentials
• Ladder operators and coherent states for the Rosen-Morse system and its rational extensions
• Generalized Version of the Creation and Annihilation Operators for the Schrödinger Equation
• Polyadic supersymmetry
• Solution of quantum eigenvalue problems by means of algebraic consistency conditions
• Nonequilibrium many-body dynamics in supersymmetric quenching
• Dynamical symmetries of supersymmetric oscillators
• Uniform magnetic field on the relativistic spinless particles with constant rest mass: 2D polar space
• Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions