A finite oscillator model related to sl(2|1)

Abstract

We investigate a new model for the finite one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). In this setting, it is natural to present the position and momentum operators of the oscillator as odd elements of the Lie superalgebra. The model involves a parameter p (0<p<1) and an integer representation label j. In the (2j+1)-dimensional representations W_j of sl(2|1), the Hamiltonian has the usual equidistant spectrum. The spectrum of the position operator is discrete and turns out to be of the form $\pm \sqrt{k}$, where k=0,1,...,j. We construct the discrete position wave functions, which are given in terms of certain Krawtchouk polynomials. These wave functions have appealing properties, as can already be seen from their plots. The model is sufficiently simple, in the sense that the corresponding discrete Fourier transform (relating position wave functions to momentum wave functions) can be constructed explicitly.

References in corpus (4)

• Realizations of $su(1,1)$ and $U_q(su(1,1))$ and generating functions for orthogonal polynomials
• The Wigner distribution function for the one-dimensional parabose oscillator
• Discrete quantum model of the harmonic oscillator
• Finite q-oscillator

Cited by in corpus (7)

• Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter
• Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)
• A superintegrable finite oscillator in two dimensions with SU(2) symmetry
• The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials
• Exact solution of the position-dependent mass Schrödinger equation with the completely positive oscillator-shaped quantum well potential
• Dynamical symmetry of a semiconfined harmonic oscillator model with a position-dependent effective mass
• The $\mathfrak{su}(2)$ Krawtchouk oscillator model under the $\mathcal{C}\mathcal{P}$ deformed symmetry