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Title: Fractional supersymmetry and hierarchy of shape invariant potentials

Abstract

Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.

Authors:
;  [1]
  1. Institut de Physique Nucleaire de Lyon, IN2P3-CNRS/Universite Claude Bernard Lyon 1, F-69622 Villeurbanne Cedex (France)
Publication Date:
OSTI Identifier:
20861554
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 12; Other Information: DOI: 10.1063/1.2401711; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; HAMILTONIANS; POTENTIALS; QUANTUM MECHANICS; SUPERSYMMETRY

Citation Formats

Daoud, M., and Kibler, M. R. Fractional supersymmetry and hierarchy of shape invariant potentials. United States: N. p., 2006. Web. doi:10.1063/1.2401711.
Daoud, M., & Kibler, M. R. Fractional supersymmetry and hierarchy of shape invariant potentials. United States. doi:10.1063/1.2401711.
Daoud, M., and Kibler, M. R. Fri . "Fractional supersymmetry and hierarchy of shape invariant potentials". United States. doi:10.1063/1.2401711.
@article{osti_20861554,
title = {Fractional supersymmetry and hierarchy of shape invariant potentials},
author = {Daoud, M. and Kibler, M. R.},
abstractNote = {Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.},
doi = {10.1063/1.2401711},
journal = {Journal of Mathematical Physics},
number = 12,
volume = 47,
place = {United States},
year = {Fri Dec 15 00:00:00 EST 2006},
month = {Fri Dec 15 00:00:00 EST 2006}
}
  • Using a supersymmetric path-integral formulation, we give explicit point canonical transformations which map the kernels (or energy-dependent Green's function) of exactly known solvable shape-invariant potentials into those of two potential classes. Exact analytic expressions for the eigenvalues and eigenfunctions of these two classes of potentials are then retrieved following a standard procedure.
  • We study the accuracy of several alternative semiclassical methods by computing analytically the energy levels for many large classes of exactly solvable shape-invariant potentials. For these potentials, the ground-state energies computed via the WKB method typically deviate from the exact results by about 10{percent}, a recently suggested modification, using nonintegral Maslov indices, is substantially better, and the supersymmetric WKB quantization method gives exact answers for all energy levels. {copyright} {ital 1997} {ital The American Physical Society}
  • We derive recursion relations for the propagators of shape-invariant potentials both in the operator formulation as well as in the path-integral formulation. This allows us to determine the form of the propagators recursively. We evaluate the propagators explicitly, for several shape-invariant potentials, using this method.
  • We develop a new approach to build the eigenfunctions of a translationally shape invariant potential. For this we show that their logarithmic derivatives can be expressed as terminating continued fractions in an appropriate variable. We give explicit formulas for all the eigenstates, their specific form depending on the Barclay-Maxwell class to which the considered potential belongs.
  • In supersymmetric quantum mechanics, shape invariance is a sufficient condition for solvability. We show that all conventional additive shape-invariant superpotentials that are independent of ({h_bar}/2{pi}) can be generated from two partial differential equations. One of these is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow, and it is closed by the other. We solve these equations, generate the set of all conventional shape-invariant superpotentials, and show that there are no others in this category. We then develop an algorithm for generating all additive shape-invariant superpotentials including those that depend on ({h_bar}/2{pi}) explicitly.