Solutions to analogues of non-stationary Schr\"odinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$

V. A. Pavlenko ; B. I. Suleimanov

Ufa mathematical journal, Tome 10 (2018) no. 4, pp. 92-102

Résumé

We construct simultaneous solutions to two analogues of time-dependent solutions to Schrödinger equations defined by the Hamiltonians $H^{2+1+1+1}_{s_k}(s_1,s_2, q_1,q_2, p_1, p_2)$ $(k=1,2)$ to system $H^{2+1+1+1}$. This system is the first representative in a famous degenerations hierarchy of the Garnier system described in 1986 by H. Kimura. By an explicit symplectic transformation, this system reduces to a symmetric Hamilton system. In the constructions of this paper we rely mostly on linear systems of equations in the method of isomonodromic deformations for the system $H^{2+1+1+1}$ written out in 2012 in a paper by A. Kavakami, A. Nakamura and H. Sakai. These analogues of the non-stationary Schrödinger equations are evolution equations with times $s_1$ and $s_2$, which depend of two spatial variables. From the canonical non-stationary Schrödinger equations, these analogues arise as a result of the formal replacement of the Planck constant by $-2\pi i$. We construct the exact solutions to the two evolution equations in terms of the solutions to corresponding linear ordinary differential equations in the method of isomonodromic deformations. We discuss further prospects for constructing similar solutions to analogues of the non-stationary Schrödinger equations corresponding to the Hamiltonians of the entire degeneracy hierarchy of the Garnier system.

Détail

Keywords: Hamilton systems, Schrödinger equation, Painlevé equations, method of isomonodromic deformations.