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.

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• Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians
• Bohm approach to the Gouy phase shift
• Supersymmetry and Shape Invariance of exceptional orthogonal polynomials
• Nonstationary deformed singular oscillator: quantum invariants and the factorization method
• Quantum control and quantum speed limits in supersymmetric potentials