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

Abstract

In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlevé and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. New quantum invariants are constructed after adding a deformation term to the well-known quantum invariant of the parametric oscillator. Such a deformation depends explicitly on time through solutions of the Ermakov equation, a property that simultaneously ensures the regularity of the new time-dependent potentials at each time. The fourth Painlevé equation appears after introducing an appropriate reparametrization of the spatial coordinate and the time parameter, where the parameters of the fourth Painlevé equation dictate the spectral information of the quantum invariant. In this form, the eigenfunctions of the third-order ladder operators lead to several sequences of solutions to the Schrödinger equation, which are determined in terms of the solutions of the Riccati equation, Okamoto polynomials, and nonlinear bound states of the derivative nonlinear Schrödinger equation. Remarkably, it is noticed that the solutions in terms of the nonlinear bound states lead to a quantum invariant with equidistant eigenvalues, which contains both an finite-dimensional and an infinite-dimensional sequences of eigenfunctions. The resulting family of time-dependent Hamiltonians is such that, to the authors’ knowledge, have been unnoticed in the literature of stationary and nonstationary systems.