“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

Abstract

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 2 1 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 2 1 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 2 1 . A similar situation occurs for the P 2 2 equation in the Painlevé II hierarchy.For all of the six Painlevé ordinary differential equations (ODEs) q zz = f (z, q, q z ), an explicit change in the solutions of the corresponding pairs of linear systems of the isomonodromic deformation method (IDM) given in Garnier's paper [1] yields solutions of the equations (see [2] and [3])(see also the beginning of the conclusion of this paper). These equations are determined by the Hamiltonians H = H(z, q, p) of Hamiltonian systems q z = H p (z, q, p) and p z = −H q (z, q, p); the Hamiltonians are quadratic in the momenta p, and eliminating p from the systems, we obtain the six Painlevé ODEs. From the Schrödinger equationswhich depend on the Planck constant h = 2π = −2πiε, the six evolution equations (1) are obtained by the formal change ε = 1. (In the cases of the Painlevé III and V equations, expressions given in [2] and [3] contain inaccuracies, which, however, are easy to correct.) Note that, for the ODEs obtained from the first and second Painlevé equations, the corresponding IDM equations can be rescaled into compatible systems of linear ODEs, which determine, for any fixed complex number ε, exact solutions of Eqs. (2) with Hamiltonians H depending on ε. Let us clarify this point. The pair of Garnier IDM equations [1] for the ODE(the a j are any constants), whose special cases are the first and second Painlevé equations, has the form W yy = P (τ, y)W, W τ = B(τ, y)W y − 1 2 B y (τ, y)W, (4) *

“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

“…Such change also played a key role in constructing solutions to "quantizations" of the Garnier system with two degrees of freedom in paper [38] and in constructing solutions to "quantizations" of two lowest representatives in the hierarchy of degenerations of this system in paper [42]. The results of these two papers and of the present paper suggest a conjecture that this change should help in constructing solutions to "quantizations" of the entire hierarchy.…”

“…It turned out [39], [40], [42] that such evolution equations are related with the representations of each of sixth canonical Painlevé ODEs ′′ = ( , , ′ ) ( = 1, . .…”

“…By excluding the momenta ( ) we get a second order ODE for the coordinate ( ) coinciding with the corresponding Painlevé equation. Another choice of order admits a symbolic writing of six linear evolution equations in form of (2) [42]. Following the terminology of paper [41], in what follows we call evolution equations (2) with constants ̸ = "quantizations" of corresponding Hamilton systems.…”

“…In particular, in works [38], [42], in terms of the corresponding solutions to the linear equations in IDM there were constructed the solutions to "quantizations" (2) for three compatibly isomonodromic Hamilton pair of systesm of ODEs with two degrees of freedom. At that, in [38] there was considered the so-called Garnier system heading an entire hierarchy of isomonodromic Hamilton systems with two degrees of freedom; these systems can be obtained from the Garnier system by the procedure of successive degeneration [11], [13].…”

“…At that, in [38] there was considered the so-called Garnier system heading an entire hierarchy of isomonodromic Hamilton systems with two degrees of freedom; these systems can be obtained from the Garnier system by the procedure of successive degeneration [11], [13]. In [42] there were constructed "quantizations" (2) of the pairs of Hamilton systems of ODEs being two lowest representatives in this hierarchy. In Remark 2 in [38] there was formulated a conjecture that by means of this procedure, constructions [38] can be extended to the entire hierarchy of the degenerations of the Garnier system.…”