Invariant partial differential equations of conformal Galilei algebra as deformations: cryptohermiticity and contractions. (2nd August 2016) - Aizawa, N.; Kuznetsova, Z.; Toppan, F.
Abstract : We investigate the general class of second-order partial differential equations (PDEs), invariant under the d = 1 ℓ = 1 2 + ℕ 0 centrally extended conformal Galilei algebras, pointing out that they are deformations of decoupled systems. For ℓ = 3 2 the unique deformation parameter γ belongs to the fundamental domain γ ∈ ] 0, + ∞ [ . We show that, for any γ ≠ 0, invariant PDEs with discrete spectra (either bounded or unbounded) induce crypto-Hermitian operators possessing the same spectrum as two decoupled oscillators, provided that their frequencies are in the special ratio r = ω 2 ω 1 = ± 1 3, ± 3 (the negative energy solutions correspond to a special case of the Pais–Uhlenbeck oscillator), where ω 1, ω 2 are two different parameters of the invariant PDEs. We also consider the γ = 0 decoupled system for any value r of the ratio. It possesses enhanced symmetry at the critical values r = ± 1 3, ± 1, ± 3 . Two inequivalent 12-generator symmetry algebras are found at r = ± 1 3, ± 3 and r = ± 1, respectively. The ℓ = 3 2 conformal Galilei algebra is not a subalgebra of the decoupled symmetry algebra. Its γ → 0 contraction corresponds to an 8-generator subalgebra of the decoupled r = ± 1 3, ± 3 symmetry algebra. The features of the ℓ ≥ 5 2 invariant PDEs are briefly discussed.