Quantum aspects of the integrability of the third Painlevé equation and a non-stationary time Schrödinger equation with the Morse potential

In terms of solutions to isomonodromic deformations equation for the third Painlevé equation, we write out the simultaneous solution of three linear partial differential equations. The first of them is a quantum analogue of the linearization of the third Painlevé equation written in one of the forms. The second is an analogue of the time Schrödinger equation determined by the Hamiltonian structure of this ordinary differential equation. The third is a first order equation with the coefficients depending explicitly on the solutions to the third Painlevé equation. For the autonomous reduction of the third Painlevé equation this simultaneous solution defines solutions to a time quantum mechanical Schrödinger equation, which is equivalent to a time Schrödinger equation with a known Morse potential. These solutions satisfy also linear differential equations with the coefficients depending explicitly on the solutions of the corresponding autonomous Hamiltonian system. It is shown that the condition of global boundedness in the spatial variable of the constructed solution to the Schrödinger equation is related to determining these solutions to the classical Hamiltonian system by Bohr–Sommerfeld rule of the old quantum mechanics.