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 Funktsional. Anal. i Prilozhen., 2014, Volume 48, Issue 3, Pages 52–62 (Mi faa3150)

“Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom

B. I. Suleimanov

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa

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 $P_1^2$ 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 $P_1^2$ 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 $P_1^2$. A similar situation occurs for the $P_2^2$ equation in the Painlevé II hierarchy.

Keywords: quantization, Schrödinger equation, Hamiltonian, Painlevé equations, isomonodromic deformations, integrability

 Funding Agency Grant Number Russian Science Foundation 14-11-00078

DOI: https://doi.org/10.4213/faa3150

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English version:
Functional Analysis and Its Applications, 2014, 48:3, 198–207

Bibliographic databases:

UDC: 517.9

Citation: B. I. Suleimanov, ““Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom”, Funktsional. Anal. i Prilozhen., 48:3 (2014), 52–62; Funct. Anal. Appl., 48:3 (2014), 198–207

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3150
• https://doi.org/10.4213/faa3150
• http://mi.mathnet.ru/eng/faa/v48/i3/p52

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. P. Novikov, B. I. Suleimanov, ““Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom”, Theoret. and Math. Phys., 187:1 (2016), 479–496
2. B. I. Suleimanov, “Quantum aspects of the integrability of the third Painlevé equation and a non-stationary time Schrödinger equation with the Morse potential”, Ufa Math. J., 8:3 (2016), 136–154
3. V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107
4. V. I. Kachalov, Yu. S. Fedorov, “O metode malogo parametra v nelineinoi matematicheskoi fizike”, Sib. elektron. matem. izv., 15 (2018), 1680–1686
5. V. A. Pavlenko, B. I. Suleimanov, “Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$”, Ufa Math. J., 10:4 (2018), 92–102
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