Nonstationary Deformed Singular Oscillator: Quantum Invariants and the Factorization Method

Abstract

Abstract New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a nonstationary quantum invariant can be constructed for the singular oscillator. Such a quantum invariant depends on coefficients related to solutions of the Ermakov equation, where the latter guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant rather than to the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators that become the constants of motion for the new time-dependent Hamiltonians. At the appropriate limit, the initial quantum invariant reproduces the stationary singular oscillator Hamiltonian. Some families of stationary potentials already reported by other authors are also recovered as particular cases. A striking feature of the method is that the singular barrier of the potential can be managed to vanish, which leads to non-singular time-dependent potentials.