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Solutions of the analogues of time-dependent Schrödinger equations corresponding to a pair of $H^{3+2}$ Hamiltonian systems
V. A. Pavlenko Institute of Mathematics with Computer Center,
Ufa Science Center, Russian Academy of Sciences, Ufa, Russia
Abstract:
We construct joint
$2\times2$ matrix solutions of the scalar linear evolution equations $\Psi'_{s_k}=H^{3+2}_{s_k}(s_1,s_2,[0]x_1,x_2, \partial/\partial x_1,\partial/\partial x_2)\Psi$ with times
$s_1$ and
$s_2$, which can be treated as analogues of the time-dependent Schrödinger equations. These equations correspond to the so-called
$H^{3+2}$ Hamiltonian system, which is a representative of a hierarchy of degenerations of the isomonodromic Garnier system described by Kimura in 1986. This compatible system of Hamiltonian ordinary differential equations is defined by two different Hamiltonians
$H^{3+2}_{s_k}(s_1,s_2,q_1,q_2,p_1,p_2)$,
$k=1,2$, with two degrees of freedom corresponding to the time variables
$s_1$ and
$s_2$. In terms of solutions of the linear systems of ordinary differential equations obtained by the isomonodromic deformation method, with the compatibility condition given by the Hamilton equations of the
$H^{3+2}$ system, the constructed compatible solutions of analogues of the time-dependent Schrödinger equations are presented explicitly. We also present a change of variables relating the matrix solutions of analogues of the time-dependent Schrödinger equations defined by two forms of the
$H^{3+2}$ system (rational and polynomial in coordinates). This system is a quantum analogue of the well-known canonical transformation relating the Hamilton equations of the
$H^{3+2}$ system in these two forms.
Keywords:
Hamiltonian systems, Painlevé-type equations, time-dependent Schrödinger equations, isomonodromic deformation method. Received: 12.03.2022
Revised: 06.05.2022
DOI:
10.4213/tmf10285
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