Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom

Abstract

For the three nonstationary Schrödinger equations i ℏ Ψ τ = H ( x , y , − i ℏ ∂ ∂ x , − i ℏ ∂ ∂ y ) Ψ , \begin{equation*} i\hbar \Psi _{\tau }=H(x,y,-i\hbar \frac {\partial }{\partial x},-i\hbar \frac {\partial }{\partial y})\Psi , \end{equation*} solutions are constructed that correspond to conservative Hamiltonian systems with two degrees of freedom whose general solutions can be represented by those of the second Painlevé equation. These solutions of the Schrödinger equations are expressed via fundamental solutions of systems of linear equations arising in the isomonodromic deformations method, the compatibility condition of which is the second Painlevé equation. The constructed solutions of two nonstationary Schrödinger equations are globally smooth. Some of the smooth solutions in question of one of these two equations exponentially tend to zero as x 2 + y 2 → ∞ x^2+y^2\to \infty if the corresponding solutions of linear systems that are used in the method of isomonodromic deformations are compatible on the so-called 1-tronquée solutions of the second Painlevé equation.