Free oscillator realization of Laguerre polynomials

We revisit the radial oscillator from the standpoint of a free oscillator realization. By using a free oscillator, namely, the creation/annihilation operators of the harmonic oscillator, we construct an operator that maps the eigenfunctions of the harmonic oscillator to those of the radial oscillator. As a polynomial part of this relation, we obtain an operator that maps the Hermite polynomials to the Laguerre polynomials.