Dynamical realization is given to the generator responsible for breaking the usual SU(3) invariance symmetry in the case of the Lewis–Riesenfeld time-dependent oscillator: H (t) = 1/2 𝒥3i[pipi+w2(t) qiqi]. This is achieved by finding a simple expression for the time-dependent Hamiltonian in the interaction picture: H=Ho+gH1. The breaking interaction is shown to transform as (6⊕6̄) of SU(3). Time-dependent dilations relate the broken Hamiltonian to the Lewis–Riesenfeld form. This scaling generates the algebra of the Weyl group and its role in oscillator noninvariance symmetries is considered.
Lewis-Riesenfeld -Ermakov's (LR) invariant method for the construction of time-dependent phase-space invariant is extended for the general quantum system with position-dependent effective mass (PDEM) Hamiltonian. It turns out that, only a specific class of PDEM and a particular class of external potentials will exhibit the LR-invariant operator in close form. Then we have determined a class of unitary time-dependent quantum canonical transformation for the concerned PDEM and external potentials so that an equivalent time-independent PDEM Hamiltonian is obtained.