Free Oscillator Realization of the Laguerre Polynomial

Abstract

Eigenfunctions of the radial oscillator are described by the Laguerre polynomial. By using a free oscillator, namely the creation/annihilation operators of the harmonic oscillator, we construct some operator. By this operator, eigenfunctions of the radial oscillator are obtained from those of the harmonic oscillator. As a polynomial part of this relation, the Laguerre polynomial L, where b is some operator and 1 F 0 is the hypergeometric function.

Free boson realization of the Dunkl intertwining operator in one dimension

“…Sticking to one parity sector, say the even one, at the price of not having a unified framework, one finds that the wave functions of the singular oscillator are obtained from those of the harmonic oscillator with an even number of excitations. This is the view taken in [15]. This observation is in keeping with the fact that on the one hand, the states of the harmonic oscillator with a fixed parity support metaplectic representations of su(1, 1) which is then mapped onto the su(1, 1) irreducible representation space spanned by the states of the singular oscillator.…”

“…Borrowing ideas presented in [15], we offer in this section a realization of the Dunkl intertwining operator V µ in terms of bosonic operators. This follows from Theorem 3.1 which gives the action of V µ on the Hermite polynomials that provide a well known representation basis for the oscillator operators.…”

“…The purpose of the present note is to offer a free field realization of the Dunkl intertwiner operator V µ in one dimension using observations made by Odake in [15] on the connection between the wave functions of the harmonic and singular oscillators. The analysis is rooted in the fact that the Hermite polynomials are mapped [8], [16], [17] onto the generalized Hermite polynomials [18], [19], [20] by the intertwining operator V µ .…”