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

Resumo

Abstract: In two papers, JPA: Math. Theor. 46, 405204 (2013) and JMP 56, 031701 (2015), second-order invariant PDEs of the $d=1$ $\ell=\frac{1}{2}+{\mathbb N}_0$ centrally extended Conformal Galilei Algebras, were constructed (for continuous and, respectively, discrete spectrum). We investigate here the general class of second-order invariant PDEs, pointing out that they are deformations of decoupled systems. For $\ell=\frac{3}{2}$ the unique
deformation parameter $\gamma$ belongs to the fundamental domain $\gamma\in ]0,+\infty[$. The invariant PDE with discrete spectrum induces a cryptohermitian operator possessing the same spectrum as two decoupled oscillators of given energy $\omega_1, \omega_2$. The normalization $\omega_1=1$ implies, for $\omega_2$, the admissible critical values $\omega_2=\pm\frac{1}{3},\pm 3$ (the negative energy solutions correspond to a special case of Pais-Uhlenbeck oscillator). \par
Unitarily inequivalent operators, acting on the ${\mathcal L}²({\mathbb R}²)$ Hilbert space, are obtained for the deformation parameter $\gamma$ belonging to the fundamental domain. The undeformed $\gamma=0$ case corresponds to a decoupled  cryptohermitian operator with enhanced symmetry at the critical values $\omega_2=\pm \frac{1}{3}, \pm 1,\pm 3$. Two inequivalent $12$-generator symmetry algebras are found at $\omega_2=\pm\frac{1}{3},\pm 3$ and $\omega_2=\pm 1$, 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 $\omega_2=\pm\frac{1}{3},\pm 3$ symmetry algebra.\par
The features of the $\ell\geq \frac{5}{2}$ invariant PDEs are briefly discussed.