Nonstationary deformed singular oscillator: quantum invariants and the factorization method

Abstract

New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a non stationary quantum invariant can be constructed for the singular oscillator. Such invariant depends on coefficients that are related to solutions of an Ermakov equation, the latter becomes essential since it guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant, rather than the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators which are the constants of motion of the new time-dependent potentials. Under the appropriate limit, the initial quantum invariant reduces to the stationary singular oscillator Hamiltonian, in such case, one recovers the families of potentials obtained through the conventional factorization method and previously reported in the literature. In addition, some special limits are discussed such that the singular barrier of the potential vanishes, leading to non-singular time-dependent potentials.

Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians

“…The latter is required to properly define the Schrödinger equation that characterize the quantum system under consideration. Such a task have been addressed in previous works using the factorization method for time-dependent Hamiltonians [56,57] by imposing the appropriate ansatz. In this work, we consider an alternative approach based on the transitionless tracking algorithm [58,62].…”

“…Lewis and Riesenfeld [52] addressed the problem by noticing the existence of a nonstationary eigenvalue equation associated with the appropriate constant of motion (quantum invariant) of the system in which the time dependence appears in the coefficients of the related ordinary differential equation. The latter eigenvalue equation can indeed be factorized in such a way that the Darboux transformation 1 is applied with ease [56,57], resulting in a new quantum invariant rather than a Hamiltonian. Then, the appropriate ansatz allows to determine the respective Hamiltonian and time-dependent potentials with ease [56].…”

“…Thus, for a fixed order N , we determine the exact form of Î1 (t). It is worth to mention that ladder operators of first and second-order have been reported for the parametric oscillator [54,56] and the nonstationary singular oscillator [57,59]. Nevertheless, in those cases, the quantum invariants are already known, and the ladder operators are constructed following the polynomial structure of the eigenfunctions.…”

“…1 for details. Now, given that Q(t) and Q † (t) are considered as first-order differential operators, we use the general form introduced in [57], that is,…”