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 Ufimsk. Mat. Zh., 2012, Volume 4, Issue 2, Pages 127–135 (Mi ufa153)

The “quantum” linearization of the Painlevé equations as a component of theier $L,A$ pairs

B. I. Suleimanov

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

Abstract: The procedure of the “quantum” linearization of the Hamiltonian ordinary differential equations with one degree of freedom is investigated. It is offered to be used for the classification of integrable equations of the Painleve type. For the Hamiltonian $H=(p^2+q^2)/2$ and all natural numbers $n$ the new solutions $\Psi(\hbar,t,x,n)$ of the non-stationary Shrödinger equation are constructed. The solutions tend to zero at $x\to\pm\infty$. On curves $x=q_n(\hbar,t)$, defined by the old Bohr–Zommerfeld rule, the solutions satisfy the relation $i\hbar\Psi'_x\equiv p_n(\hbar,t)\Psi$. In this relation $p_n(\hbar,t)=(q_n(\hbar,t))'_t$ is the classical momentum corresponding to the harmonic $q_n(\hbar,t)$.

Keywords: quantization, linearization, non-stationary Schrödinger equation, Painlevé equations, isomonodromi deformations.

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Citation: B. I. Suleimanov, “The “quantum” linearization of the Painlevé equations as a component of theier $L,A$ pairs”, Ufimsk. Mat. Zh., 4:2 (2012), 127–135

Citation in format AMSBIB
\Bibitem{Sul12} \by B.~I.~Suleimanov \paper The quantum'' linearization of the Painlev\'e equations as a~component of theier $L,A$ pairs \jour Ufimsk. Mat. Zh. \yr 2012 \vol 4 \issue 2 \pages 127--135 \mathnet{http://mi.mathnet.ru/ufa153} 

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This publication is cited in the following articles:
1. B. I. Suleimanov, ““Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom”, Funct. Anal. Appl., 48:3 (2014), 198–207
2. H. Rosengren, “Special Polynomials Related to the Supersymmetric Eight-Vertex Model: a Summary”, Commun. Math. Phys., 340:3 (2015), 1143–1170
3. 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
4. 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
5. I. T. Habibullin, A. R. Khakimova, “Invariant manifolds and Lax pairs for integrable nonlinear chains”, Theoret. and Math. Phys., 191:3 (2017), 793–810
6. V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107
7. R. Conte, “Generalized Bonnet Surfaces and Lax Pairs of P$_{\mathrm{VI}}$”, J. Math. Phys., 58:10 (2017), 103508
8. I. T. Habibullin, A. R. Khakimova, “On a Method For Constructing the Lax Pairs For Integrable Models Via a Quadratic Ansatz”, J. Phys. A-Math. Theor., 50:30 (2017), 305206
9. I. T. Habibullin, A. R. Khakimova, “A direct algorithm for constructing recursion operators and Lax pairs for integrable models”, Theoret. and Math. Phys., 196:2 (2018), 1200–1216
10. 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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