``Quantizations'' of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom (Q2258219)

In [\textit{B. I. Suleimanov}, ``Hamiltonian property of the Painlevé equations and the method of isomonodromic deformations'', Differ. Equations 30, No. 5, 726--732 (1994); translation from Differ. Uravn. 30, No. 5, 791--796 (1994; Zbl 0844.34012)] there was obtained a quantization of Painlevé equations. By this, the author means the following construction. As it is known, using the isomonodromy deformation method one can write a Hamiltonian system equivalent to a Painlevé equation. More precise, there exists a Hamiltonian system with a non-autonomous Hamiltonian function \(H(z,p,q)\) such that if one eliminates \(p\) from the the Hamiltonian system one obtaines a Painlevé equation. In the paper cited above the author takes the Schrödinger equation with an arbitrary Planck constant corresponding to this Hamiltonian and obtaines a ``quantization'' of the Painlevé equation. Also the author obtains solutions of this quantized equation. In the present paper, an analogous constrution is done for the second equations in Painlevé I and Painlevé II hierarchies. Solutions of the quantizations are obtained. One should remark that in this case the Hamiltonian depends on two times \(z\) and \(t\).