$\ell$-oscillators from second-order invariant PDEs of the centrally extended Conformal Galilei Algebras

Abstract

We construct, for any given $\ell =\frac{1}{2}+{\mathbb{N}}_0$, the second-order, linear PDEs which are invariant under the centrally extended Conformal Galilei Algebra. \par At the given $\ell$, two invariant equations in one time and $\ell +\frac{1}{2}$ space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrödinger equation (recovered for $\ell =\frac{1}{2}$) in $1+1$ dimension. The second equation (the "$\ell$-oscillator") possesses a discrete, positive spectrum. It generalizes the $1+1$-dimensional harmonic oscillator (recovered for $\ell =\frac{1}{2}$). The spectrum of the $\ell$-oscillator, derived from a specific $osp(1|2\ell +1)$ h.w.r., is explicitly presented.\par The two sets of invariant PDEs are determined by imposing (representation-dependent) {\it on-shell invariant conditions} both for {\it degree} $1$ operators (those with continuum spectrum) and for {\it degree } $0$ operators (those with discrete spectrum).\par The on-shell condition is better understood by enlarging the Conformal Galilei Algebras with the addition of certain second-order differential operators. Two compatible structures (the algebra/superalgebra duality) are defined for the enlarged set of operators.

arXiv comment18 pages

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