Invariant PDEs of Conformal Galilei Algebra as deformations: cryptohermiticity and contractions

Abstract

We investigate the general class of second-order PDEs, invariant under the $d=1$ $\ell=\frac{1}{2}+{\mathbb N}_0$ centrally extended Conformal Galilei Algebras, pointing out that they are deformations of decoupled systems. For $\ell=\frac{3}{2}$ the unique deformation parameter $\gamma$ belongs to the fundamental domain $\gamma\in ]0,+\infty[$. We show that, for any $\gamma\neq 0$, invariant PDEs with discrete spectrum (either bounded or unbounded) induce cryptohermitian operators possessing the same spectrum as two decoupled oscillators, provided that their frequencies are in the special ratio $r=\frac{\omega_2}{\omega_1}=\pm\frac{1}{3},\pm 3$ (the negative energy solutions correspond to a special case of Pais-Uhlenbeck oscillator), where $\omega_1,\omega_2$ are two different parameters of the invariant PDEs. We also consider the $\gamma=0$ decoupled system for any value $r$ of the ratio. It possesses enhanced symmetry at the critical values $r=\pm \frac{1}{3}, \pm 1,\pm 3$. Two inequivalent $12$-generator symmetry algebras are found at $r =\pm\frac{1}{3},\pm 3$ and $r=\pm 1$, respectively. The $\ell=\frac{3}{2}$ Conformal Galilei Algebra is not a subalgebra of the decoupled symmetry algebra. Its $\gamma\rightarrow 0$ contraction corresponds to a $8$-generator subalgebra of the decoupled $r=\pm\frac{1}{3},\pm 3$ symmetry algebra. The features of the $\ell\geq \frac{5}{2}$ invariant PDEs are briefly discussed.