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

This article is cited in 10 scientific papers (total in 10 papers)

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.

Full text: PDF file (469 kB)
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UDC: 517.9
Received: 01.03.2012

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. H. Rosengren, “Special Polynomials Related to the Supersymmetric Eight-Vertex Model: a Summary”, Commun. Math. Phys., 340:3 (2015), 1143–1170  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  isi  elib
    7. R. Conte, “Generalized Bonnet Surfaces and Lax Pairs of P$_{\mathrm{VI}}$”, J. Math. Phys., 58:10 (2017), 103508  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    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  mathnet  crossref  isi
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