Solutions of the~analogues of time-dependent Schr\"odinger equations corresponding to a~pair of $H^{3+2}$ Hamiltonian systems

V. A. Pavlenko

Teoretičeskaâ i matematičeskaâ fizika, Tome 212 (2022) no. 3, pp. 340-353

Résumé

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.

Détail

Keywords: Hamiltonian systems, Painlevé-type equations, time-dependent Schrödinger equations
Mots-clés : isomonodromic deformation method.