Abstract: 
In this talk, I discuss a new realization of exactly solvable time-dependent Hamiltonians based on the solutions of the fourth Painleve 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.  The new quantum invariant is constructed by adding a deformation term to the well-known parametric oscillator invariant.  Such a deformation depends explicitly on time through the solutions of the Ermakov equation, which ensures the regularity of the new time-dependent potential of the Hamiltonian at each time.  The fourth Painleve equation appears naturally with the aid of the proper reparametrization, whose parameters dictate the form of the discrete spectrum of the quantum invariant.  Some particular examples are presented to illustrate the results.
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Joint work with Ian Marquete and Veronique Hussin.