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Solutions of the analogues of time-dependent Schrödinger equations corresponding to a pair of \(H^{3+2}\) Hamiltonian systems

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Abstract

We construct joint \(2\times2\) matrix solutions of the scalar linear evolution equations \(\Psi'_{s_k}=H^{3+2}_{s_k}(s_1,s_2, x_1,x_2, \partial/\partial x_1,\partial/\partial x_2)\Psi\) 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.

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  1. Institute of Mathematics with Computer Center, Ufa Science Center, Russian Academy of Sciences, Ufa, Russia

    V. A. Pavlenko

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 212, pp. 340–353 https://doi.org/10.4213/tmf10285.

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Pavlenko, V.A. Solutions of the analogues of time-dependent Schrödinger equations corresponding to a pair of \(H^{3+2}\) Hamiltonian systems. Theor Math Phys 212, 1181–1192 (2022). https://doi.org/10.1134/S0040577922090021

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