`-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

We construct, for any given ` = 12 + N0, the second-order, linear PDEs which are invariant under the centrally extended Conformal Galilei Algebra. At the given `, two invariant equations in one time and ` + 12 space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrödinger equation (recovered for ` = 12) in 1 + 1 dimension. The second equation (the “`-oscillator”) possesses a discrete, positive spectrum. It generalizes the 1 + 1-dimensional harmonic oscillator (recovered for ` = 12). The spectrum of the `-oscillator, derived from a specific osp(1|2` + 1) h.w.r., is explicitly presented. The two sets of invariant PDEs are determined by imposing (representationdependent) on-shell invariant conditions both for degree 1 operators (those with continuum spectrum) and for degree 0 operators (those with discrete spectrum). 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. ∗The work has been accepted in Journal of Mathematical Physics. †E-mail: [email protected] ‡E-mail: [email protected] §E-mail: [email protected]