Geodesic
``Quantizations'' of Higher Hamiltonian Analogues of the Painlev\'e I and Painlev\'e II Equations with Two Degrees of Freedom
Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 3, pp. 52-62

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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.
Mots-clés : quantization, isomonodromic deformations
Keywords: Schrödinger equation, Hamiltonian, Painlevé equations, integrability. 
@article{FAA_2014_48_3_a4,
     author = {B. I. Suleimanov},
     title = {``Quantizations'' of {Higher} {Hamiltonian} {Analogues} of the {Painlev\'e} {I} and {Painlev\'e} {II} {Equations} with {Two} {Degrees} of {Freedom}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {52--62},
     publisher = {mathdoc},
     volume = {48},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2014_48_3_a4/}
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B. I. Suleimanov. ``Quantizations'' of Higher Hamiltonian Analogues of the Painlev\'e I and Painlev\'e II Equations with Two Degrees of Freedom. Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 3, pp. 52-62. http://geodesic.mathdoc.fr/item/FAA_2014_48_3_a4/