Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method

Abstract

New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of the parametric oscillator, leading to new families of quantum invariants that are almost-isospectral to the initial one. Then, the respective time-dependent Hamiltonians are constructed, and the solutions of the Schrödinger equation are determined from the intertwining relationships and by finding the appropriate time-dependent complex-phases of the Lewis-Riesenfeld approach. To illustrate the results, the set of parameters of the new potentials are fixed such that a family of time-dependent rational extensions of the parametric oscillator is obtained. Moreover, the rational extensions of the harmonic oscillator are recovered in the appropriate limit.

Bohm approach to the Gouy phase shift

“…(5) that is a solution of equation ( 4); with , , are arbitrary real numbers, with the constraint − 2 ∕4 = 1. It is worth to mention that for the singular oscillator with constant frequency, there also exists a generalization of the Ermakov invariant which is also explicitly time dependent [33,34,35].…”

Exact solutions for time-dependent non-Hermitian oscillators: classical and quantum pictures

“…In turn, the parametric oscillator is found to have a constant of motion that defines the appropriate eigenvalue equation [23]. The latter result motivated the systematic research of quantum invariants [25][26][27][28], with applications in the construction of time-dependent wave-packets [29][30][31][32][33][34][35][36][37], Darboux transformations [38][39][40][41], and two-dimensional photonic systems [42], among other. The Swanson oscillator [43] is a very peculiar system that combines both profiles since it is non-Hermitian and time-dependent.…”

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

“…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].…”

“…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]. The solutions, and the complex-phases introduced by Lewis-Riesenfeld, are inherited from the former system, ensuring an orthogonal set of solutions for the new system.…”

“…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.…”