“Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom

We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian $ H_1 (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painlevé I hierarchy. This solution is obtained by an explicit change of variables from a solution of systems of linear equations whose compatibility condition is the ordinary differential equation $P_1^2$ with respect to $z$. This solution also satisfies an analogue of the Schrödinger equation corresponding to the Hamiltonian $ H_2 (z, t, q_1, q_2, p_1, p_2) $ of a Hamiltonian system with respect to $t$ compatible with $P_1^2$. A similar situation occurs for the $P_2^2$ equation in the Painlevé II hierarchy.