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 $\gamma \ne 0$, 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 $\gamma \to 0$ contraction corresponds to a $8$-generator subalgebra of the decoupled $r=\pm \frac{1}{3},\pm 3$ symmetry algebra. The features of the $\ell \ge \frac{5}{2}$ invariant PDEs are briefly discussed.

18 pages; final version to appear in PTEP

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Cited by in corpus (3)

• Schrödinger symmetry: a historical review
• Nonlocal dynamics and infinite non-relativistic conformal symmetries
• Aspects of infinite dimensional $\ell$-super Galilean conformal algebra