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

TL;DR

This paper studies second-order PDEs invariant under conformal Galilei algebra, revealing their deformation structure, cryptohermiticity, and special symmetry properties at critical parameter ratios, with implications for oscillator spectra.

Contribution

It characterizes invariant PDEs as deformations of decoupled systems, analyzes cryptohermiticity, and identifies symmetry enhancements at specific parameter ratios for the conformal Galilei algebra.

Findings

Invariant PDEs induce cryptohermitian operators with oscillator spectra.

Special ratios of parameters lead to symmetry enhancement and critical behavior.

The $ ext{Conformal Galilei Algebra}$ is not a subalgebra of the decoupled symmetry algebra.

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.