Nonlinear Supersymmetry for Spectral Design in Quantum Mechanics

Abstract

Nonlinear (Polynomial, N-fold) SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Possible extensions of SUSY in one dimension are described. They include (no more than) $\mathcal{N}=2$ extended SUSY with two nilpotent SUSY charges which generate the hidden symmetry acting as a central charge. Embedding stationary quantum systems into a non-stationary SUSY QM is shown to yield new insight on quantum orbits and on spectrum generating algebras.

30 pages, IOP style, talk at PSQM-2003 (Valladolid); minor corrections in refs

References in corpus (64)

• Supersymmetry and Quantum Mechanics
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Higher-Derivative Supersymmetry and the Witten Index
• SUSY Quantum Mechanics with Complex Superpotentials and Real Energy Spectra
• Pseudo-Supersymmetric Quantum Mechanics and Isospectral Pseudo-Hermi tian Hamiltonians
• Second Order Derivative Supersymmetry and Scattering Problem
• Supersymmetry without hermiticity
• Polynomial SUSY in Quantum Mechanics and Second Derivative Darboux Transformation
• Universal Superpositions of Coherent States and Self-Similar Potentials
• N-fold Supersymmetry in Quantum Mechanics - General Formalism -
• Bound States in the Continuum from Supersymmetric Quantum Mechanics
• Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
• The finite difference algorithm for higher order supersymmetry
• Nonlinear supersymmetry in Quantum Mechanics: algebraic properties and differential representation
• Systems with Higher-Order Shape Invariance: Spectral and Algebraic Properties
• New hydrogen-like potentials
• General Forms of a N-fold Supersymmetric Family
• Nonlinear Supersymmetry, Quantum Anomaly and Quasi-Exactly Solvable Systems
• New possibilities for supersymmetry breakdown in quantum mechanics and second order irreducible Darboux transformations
• SUSUSY quantum mechanics
• New Methods for Two-Dimensional Schrödinger Equation: SUSY-separation of Variables and Shape Invariance
• New Solvable and Quasi Exactly Solvable Periodic Potentials
• New features in supersymmetry breakdown in quantum mechanics
• The confluent algorithm in second order supersymmetric quantum mechanics
• Group theoretical approach to the intertwined Hamiltonians
• Exactly Solvable Hydrogen-like Potentials and Factorization Method
• Matrix Hamiltonians: SUSY approach to hidden symmetries
• Second order SUSY transformations with `complex energies'
• Dyonic black holes in effective string theories
• SUSY-Approach for Investigation of Two-Dimensional Quantum Mechanical Systems
• Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra
• Valley Views: Instantons, Large Order Behaviors, and Supersymmetry
• Weak supersymmetry
• Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
• Algebraic Solution of the Supersymmetric Hydrogen Atom in d Dimensions
• Two-dimensional SUSY-pseudo-Hermiticity without separation of variables
• The Phenomenon of Darboux Displacements
• SUSY Intertwining Relations of Third Order in Derivatives
• The Quantum Mechanics SUSY Algebra: An Introductory Review
• Intertwining relations of non-stationary Schrödinger operators
• Superconformal mechanics and nonlinear supersymmetry
• Erratum: Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians [J. Math. Phys. 43, 6343 (2002); math-ph/0207009]
• On form-preserving transformations for the time-dependent Schrödinger equation
• Non-Hermitian SUSY Hydrogen-like Hamiltonians with real spectra
• New Families of Isospectral Hydrogen-like potentials
• Classical Integrable 2-dim Models Inspired by SUSY Quantum Mechanics
• Interaction via reduction and nonlinear superconformal symmetry
• Phase equivalent chains of Darboux transformations in scattering theory
• Superposition effect and clan structure in forward-backward multiplicity correlations
• sl(2) Construction of Type A N-fold supersymmetry
• Supersymmetric quantum mechanics based on higher excited states
• N-fold Supersymmetry in Quantum Mechanics - Analyses of Particular Models -
• Nonlinear Holomorphic Supersymmetry on Riemann Surfaces
• Quantum mechanical spectral engineering by scaling intertwining
• Classification of Type A N-fold supersymmetry
• Higher-Order Derivative Susy in Quantum Mechanics with Large Energy Shifts
• Nonlinear supersymmetry: from classical to quantum mechanics
• Darboux transformations for quasi-exactly solvable Hamiltonians
• Higher Order SUSY in Quantum Mechanics and Integrability of Two-dimensional Hamiltonians
• A Family of Quasi-solvable Quantum Many-body Systems
• Nilpotent Marsh and SUSY QM
• Global Symmetries of Time-Dependent Schrodinger Equations
• Polynomial Heisenberg algebras and higher order supersymmetry
• On Time-Dependant Symmetries of Schroedinger Equation

Cited by in corpus (62)

• Coherent states for Hamiltonians generated by supersymmetry
• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• Position Dependent Mass Oscillators and Coherent States
• Extended SUSY quantum mechanics, intertwining operators and coherent states
• Non-Hermitian Quantum Mechanics of Non-diagonalizable Hamiltonians: puzzles with self-orthogonal states
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• New supersymmetry-generated complex potentials with real spectra
• Wronskian formula for confluent second-order supersymmetric quantum mechanics
• Factorization method and new potentials from the inverted oscillator
• Supersymmetric partners of the trigonometric Poschl-Teller potentials
• Trends in supersymmetric quantum mechanics
• Generalized coherent states for time-dependent and nonlinear Hamiltonians via complex Riccati equations
• Wronskian differential formula for confluent supersymmetric quantum mechanics
• New Two-Dimensional Quantum Models Partially Solvable by Supersymmetrical Approach
• Spectrally isomorphic Dirac systems: graphene in electromagnetic field
• On integral and differential representations of Jordan chains and the confluent supersymmetry algorithm
• 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
• Confluent Crum-Darboux transformations in Dirac Hamiltonians with $PT$-symmetric Bragg gratings
• Supercharge optical arrays
• Exactly solvable associated Lame potentials and supersymmetric transformations
• Construction of PT-asymmetric non-Hermitian Hamiltonians with CPT-symmetry
• CPT-conserving Hamiltonians and their nonlinear supersymmetrization using differential charge-operators C
• New supersymmetric partners for the associated Lame potentials
• Recursive Representation of Wronskians in Confluent Supersymmetric Quantum Mechanics
• Coherent states for exactly solvable time-dependent oscillators generated by Darboux transformations
• Hyperconfluent third-order supersymmetric quantum mechanics
• SU(1,1) Coherent States For Position-Dependent Mass Singular Oscillators
• Double Shape Invariance of Two-Dimensional Singular Morse Model
• Irreducible second order SUSY transformations between real and complex potentials
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: II. Rigorous results
• Two-Dimensional Supersymmetry: From SUSY Quantum Mechanics to Integrable Classical Models
• Metric versus observable operator representation, higher spin models
• Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model
• Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons
• Complex oscillator and Painlevé IV equation
• Factorization of nonlinear supersymmetry in one-dimensional Quantum Mechanics. I: general classification of reducibility and analysis of the third-order algebra
• Exact propagators for SUSY partners
• Confluent Second-Order Supersymmetric Quantum Mechanics and Spectral Design
• Analytical Solution of Two-Dimensional Scarf II Model by Means of SUSY Methods
• New Exactly Solvable Isospectral Partners for PT Symmetric Potentials
• Rectangular Potentials in a Semi-Harmonic Background: Spectrum, Resonances and Dwell Time
• Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics
• Spectral Design for Matrix Hamiltonians: Different Methods of Constructing of a Matrix Intertwining Operator
• Polynomial Supersymmetry for Matrix Hamiltonians
• Minimal Realizations of Supersymmetry for Matrix Hamiltonians
• Supersymmetrical Separation of Variables for Scarf II Model: Partial Solvability
• Factorization of non-linear supersymmetry in one-dimensional Quantum Mechanics. II: proofs of theorems on reducibility
• Shape invariance through Crum transformation
• Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
• Extended WKB method, resonances and supersymmetric radial barriers
• Higher Order Supersymmetric Truncated Oscillators
• Supersymmetry Transformations for Delta Potentials
• Hallmarks of tunneling dynamics with broken reflective symmetry
• Quasi-exact solvability and intertwining relations
• Harmonic Oscillator SUSY Partners and Evolution Loops
• Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics
• SIP-potentials and self-similar potentials of Shabat and Spiridonov: space asymmetric deformation
• Freezable bound states in the continuum for time-dependent quantum potentials
• Search of a general form of superpotential in hierarchy with discrete energy spectrum
• Invisible nuclear system
• Polynomial supersymmetry for matrix Hamiltonians: proofs