Geodesic
“Quantizations” of the second Painlevé equation and the problem of
Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 3, pp. 364-377 Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that the known Flaschka–Newell $L$–$A$ pair for the second Painlevé equation gives solutions to linear evolutionary equations similar to the quantum Schrödinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.
Keywords: quantum Schrödinger equation, Hamiltonian
Mots-clés : Painlevé equations, isomonodromic deformation. 
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B. I. Suleimanov. “Quantizations” of the second Painlevé equation and the problem of. Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 3, pp. 364-377. http://geodesic.mathdoc.fr/item/TMF_2008_156_3_a3/

[1] H. Flaschka, J. Math. Phys., 21:5 (1981), 1016–1018 | DOI | MR | Zbl

[2] S. P. Novikov, Funkts. analiz i ego pril., 24:4 (1990), 43–53 | MR | Zbl

[3] I. M. Krichever, On Heisenberg relations for the ordinary linear differential operators, Preprint IHES, Bur-Sur-Yvette, 1990

[4] R. Garnier, Ann. Sci. École Norm. Sup. (4), 29 (1912), 1–126 | DOI | MR | Zbl

[5] B. I. Suleimanov, “Gamiltonova struktura uravnenii Penleve i metod izomonodromnykh deformatsii”, Asimptoticheskie svoistva reshenii differentsialnykh uravnenii, In-t matematiki, Ufa, 1988, 93 ; Дифференц. уравнения, 30:5 (1994), 791–796 | MR | MR | Zbl

[6] R. Fuchs, Math. Ann., 63:3 (1907), 301–321 | DOI | MR | Zbl

[7] A. I. Ovseevich, Dokl. RAN, 414:6 (2007), 732–735 | Zbl

[8] A. A. Kapaev, E. Hubert, J. Phys. A, 32 (1999), 8145–8156 | DOI | MR | Zbl

[9] A. A. Kapaev, “Lax pairs for Painlevé equations”, Isomonodromic Deformations and Applications in Physics (Montréal, Canada, May 1–6, 2000), CRM Proc. Lecture Notes, 31, AMS, Providence, RI, 2002, 37–48 | DOI | MR | Zbl

[10] H. Flashka, A. Newell, Comm. Math. Phys., 76:1 (1980), 65–116 | DOI | MR | Zbl

[11] F. R. Gantmakher, Lektsii po analiticheskoi mekhanike, Nauka, M., 1966 | MR | Zbl

[12] A. Messia, Kvantovaya mekhanika, t. 1, Nauka, M., 1978 | MR | MR | Zbl

[13] K. Okamoto, Proc. Japan Acad. Ser. A Math. Sci., 56:6 (1980), 264–268 | DOI | MR | Zbl

[14] K. Okamoto, “The Hamiltonian associated to the Painlevé equations”, The Painleve Property. One Century Later, CRM Ser. Math. Phys., Springer, New York, 1999, 735–787 | MR | Zbl

[15] A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, A. S. Fokas, Transtsendenty Penleve. Metod zadachi Rimana, RKhD, Moskva–Izhevsk, 2005 | MR | Zbl

[16] Y. Ohyama, S. Okumura, J. Phys. A, 39:39 (2006), 12129–12151 | DOI | MR | Zbl

[17] K. Okamoto, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 33:3 (1986), 575–618 | MR | Zbl

[18] J. Harnad, Comm. Math. Phys., 166:2 (1994), 337–365 | DOI | MR | Zbl

[19] J. Harnad, A. R. Its, Comm. Math. Phys., 226:3 (2002), 497–530 | DOI | MR | Zbl

[20] B. Dubrovin, “Painlevé transcendents in two dimensional topological field theory”, The Painleve Property. One Century Later, CRM Ser. Math. Phys., Springer, New York, 1999, 287–412 | MR | Zbl

[21] M. Mazzocco, “Painlevé sixth equation as isomonodromic deformations equation of an irregular system”, The Kowalevski Property (Leeds, 2000), CRM Proc. Lecture Notes, 32, AMS, Providence, RI, 2002, 219–238 | DOI | MR | Zbl

[22] N. Joshi, A. V. Kitaev, P. A. Treharne, J. Math. Phys., 48:10 (2007), 103512 ; arXiv: 0706.1750 | DOI | MR | Zbl

[23] G. Bremerman, Raspredeleniya, kompleksnye peremennye i preobrazovaniya Fure, Mir, M., 1968 | MR | MR | Zbl

[24] G. E. Shilov, Matematicheskii analiz. Vtoroi spetsialnyi kurs, Nauka, M., 1965 | MR | MR

[25] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii i deistviya nad nimi, Vyp. 1, Fizmatlit, M., 1959 | MR | Zbl

[26] C. P. Balandin, B. I. Suleimanov, Differents. uravneniya, 30:12 (1994), 2175–2176 | MR | Zbl

[27] W. Heisenberg, Z. Phys., 33:1 (1925), 879–893 | DOI | Zbl

[28] I. Yu. Kobzarev, Yu. I. Manin, Elementarnye chastitsy: dialogi fizika i matematika, Fazis, M., 1997 | MR | MR | Zbl

[29] S. Yu. Slavyanov, J. Phys. A, 29:22 (1996), 7329–7335 | DOI | MR | Zbl

[30] C. Yu. Slavyanov, TMF, 119:1 (1999), 3–19 | DOI | MR | Zbl

[31] C. Yu. Slavyanov, TMF, 123:3 (2000), 395–406 | DOI | MR | Zbl

[32] S. Yu. Slavyanov, V. Lai, Spetsialnye funktsii. Edinaya teoriya, osnovannaya na analize osobennostei, Nevskii dialekt, S.-Pb., 2002 | MR | Zbl

[33] C. Yu. Slavyanov, F. R. Vuyalkovich, TMF, 150:1 (2007), 143–151 | DOI | MR | Zbl

[34] A. V. Zhiber, A. B. Shabat, Dokl. AN SSSR, 247:5 (1979), 1103–1107 | MR

[35] C. I. Svinolupov, UMN, 40:5(245) (1985), 263–264 | MR | Zbl

[36] A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, Comm. Math. Phys., 115:1 (1988), 1–19 | DOI | MR | Zbl

[37] A. V. Mikhailov, A. B. Shabat, V. V. Sokolov, “The symmetry approach to classification of integrable equations”, What is Integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 115–184 | DOI | MR | Zbl

[38] V. E. Adler, A. B. Shabat, R. I. Yamilov, TMF, 125:3 (2000), 355–424 | DOI | MR | Zbl

[39] A. V. Zhiber, V. V. Sokolov, UMN, 56:1 (2001), 63–106 | DOI | MR | Zbl

[40] R. Yamilov, J. Phys. A, 39:45 (2006), R541–R623 | DOI | MR | Zbl