Isomonodromic quantization of the second Painlev\'e equation by means of conservative Hamiltonian systems with two degrees of freedom

B. I. Suleimanov

Algebra i analiz, Tome 33 (2021) no. 6, pp. 141-161.

Bibliographie

[1] Bloemendal A., Virag B., “Limits of spiked random matrices I”, Probability Theory Related Fields, 156:3-4 (2013), 795–825

[2] Bloemendal A., Virag B., “Limits of spiked random matrices II”, Ann. Probab., 44:4 (2016), 2726–2769

[3] Boutroux P., “Researches sur les transcendates de M. Painlevé et létude asymptotique des equations différentielles du second ordre”, Ann. Sci. Ecole Norm. Sup. (3), 30 (1913), 255–375

[4] Boutroux P., “Researches sur les transcendates de M. Painlevé et létude asymptotique des equations différentielles du second ordre (suite)”, Ann. Sci. Ecole Norm. Sup. (3), 31 (1914), 99–159

[5] Conte R., “Generalized Bonnet surfaces and Lax pairs of PVI”, J. Math. Phys., 2017, no. 10, 103508

[6] Conte R., Dornic I., “The master Painlevé VI heat equation”, C. R. Math. Acad. Sci. Paris, 352:10 (2014), 803–806

[7] Flaschka H., Newell A., “Monodromy- and spectrum-preserving deformations $I$”, Comm. Math. Phys., 76:1 (1980), 65–116

[8] Garnier R., “Sur des equations différentielles du troisieme ordre dont l'integrale generale est uniforme et sur une classe d'equations nouvelles d'ordre superieur dont l'integrale generale a ses points critiques fixes”, Ann. Sci. Ecole Norm. Sup. (3), 29 (1912), 1–126

[9] Grava T., Its A., Kapaev A., Mezzadri F., “On the Tracy-Widom$_\beta$ distribution for $\beta = 6$”, SIGMA Symmetry Integrability Geom. Methods Appl., 12 (2016), 105

[10] Grundland A. M., Riglioni D., “Classical-quantum correspondence for shape-invariant systems”, J. Phys. A, 48:24 (2015), 245201–245215

[11] Its A. R., Novokshenov V. Yu., The isomonodromic deformation method in the theory of Painlevé equations, Lecture Notes in Math., 1191, Springer-Verlag, Berlin, 1986

[12] Levin A. M., Olshanetsky M. A., Zotov A. V., “Planck constant as spectral parameter in integrable systems and KZB equations”, J. High Energy Phys., 10 (2014), 109

[13] Miller P. D., “On the increasing tritronquée solutions of the Painlevé-II equation”, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), 125, 38 pp.

[14] Nagoya H., “Hypergeometric solutions to Schrödinger equation for the quantum Painlevé equations”, J. Math. Phys., 52:8 (2011), 083509

[15] Nagoya N., Yamada Y., “Symmetries of quantum Lax equations for the Painlevé equations”, Ann. Henri Poincaré, 15:2 (2014), 313–344

[16] Novikov D. P., “A monodromy problem and some functions connected with Painlevé $6$”, Proc. Inter. Conf. Painleve Equations and Related Topics, Euler Internat. Math. Inst., St.-Petrsburg, 2011, 118–121

[17] Rosengren H., Special polynomials related to the supersymmetric eight-vertex model. II. Schrödinger equation, 2013, arXiv: 1312.5879

[18] Rosengren H., “Special polynomials related to the supersymmetric eight-vertex model: a summary”, Comm. Math. Phys., 15:3 (2015), 1143–1170

[19] Rumanov I., “Hard edge for beta-ensembles and Painleve III”, Int. Math. Res. Not. IMRN, 23, 2014, 6576–6617

[20] Rumanov I., “Classical integrability for beta-ensembles and general Fokker–Planck equations”, J. Math. Phys., 56:1 (2015), 013508

[21] Rumanov I., “Beta ensembles, quantum Painlevé equations and isomonodromy systems”, Algebraic and analytic aspects of integrable systems and Painleve equations, Contemp. Math., 651, Amer. Math. Soc., Providence, RI, 2015, 125–155

[22] Rumanov I., “Painlevé representation of Tracy–Widom$_\beta$ distribution for $\beta = 6$”, Comm. Math. Phys., 342:3 (2016), 843–868

[23] Zabrodin A., Zotov A., “Quantum Painlevé-Calogero correspondence”, J. Math. Phys., 53:7 (2012), 073507

[24] Zabrodin A., Zotov A., “Classical-quantum correspondence and functional relations for Painlevé equations”, Constr. Approx., 41:3 (2015), 385–423

[25] Vazov V., Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1968

[26] Zotov A. V., Smirnov A. V., “Modifikatsiya rassloenii, ellipticheskie integriruemye sistemy i svyazannye zadachi”, Teor. i mat. fiz., 177:1 (2013), 3–67

[27] Its A. R., Kapaev A. A., Novokshenov V. Yu., Fokas A. S., Transtsendenty Penleve. Metod zadachi Rimana, In-t komp. issled.; NITs Reg. i khaotich. dinamika, M.–Izhevsk, 2005

[28] Kitaev A. V., “Tochki povorota lineinykh sistem i dvoinye asimptotiki transtsendentov Penleve”, Zap. nauch. semin. LOMI, 187, 1991, 53–74

[29] Levin A. M., Olshanetskii M. A., Zotov A. V., “Klassifikatsiya izomonodromnykh zadach na ellipticheskikh krivykh”, Uspekhi mat. nauk, 69:1 (2014), 39–124

[30] Messia A., Kvantovaya mekhanika, v. 1, Nauka, M., 1978

[31] Novikov D. P., “O sisteme Shlezingera s matritsami razmera $2\times2$ i uravnenii Belavina–Polyakova–Zamolodchikova”, Teor. i mat. fiz., 161:2 (2009), 2191–203

[32] Novikov D. P., Romanovskii R. K., Sadovnichuk S. G., Nekotorye novye metody konechnozonnogo integrirovaniya solitonnykh uravnenii, Nauka, Novosibirsk, 2013

[33] Novikov D. P., Suleimanov B. I., ““Kvantovaniya” izomonodromnoi gamiltonovoi sistemy Garne s dvumya stepenyami svobody”, Teor. i mat. fiz., 187:1 (2016), 39–57

[34] Novokshenov V. Yu., “Usechennye resheniya uravneniya Penleve II”, Teor. i mat. fiz., 172:2 (2012), 296–307

[35] Novokshenov V. Yu., “Spetsialnye resheniya pervogo i vtorogo uravnenii Penleve i osobennosti mnogoobraziya dannykh monodromii”, Tr. IMM UrO RAN, 18, no. 2, 2012, 179–190

[36] Pavlenko V. A., Suleimanov B. I., ““Kvantovaniya” izomonodromnykh gamiltonovykh sistem $H^{\frac{7}{2}+1}$”, Ufimsk. mat. zh., 9:4 (2017), 100–110

[37] Pavlenko V. A., Suleimanov B. I., “Resheniya analogov vremennykh uravnenii Shredingera, opredelyaemykh izomonodromnoi gamiltonovoi sistemoi $H^{2+1+1+1}$”, Ufimsk. mat. zh., 10:4 (2018), 92–102

[38] Pavlenko V. A., Suleimanov B. I., “Yavnye resheniya analogov vremennykh uravnenii Shredingera s gamiltonovoi sistemoi $H^{4+1}$”, Izv. RAN. Ser. fiz., 84:5 (2020), 695–698

[39] Culeimanov B. I., “Gamiltonova struktura uravnenii Penleve i metod izomonodromnykh deformatsii”, Asimptoticheskie svoistva reshenii differentsialnykh uravnenii, I-nt mat., Ufa, 1988, 93–102

[40] Culeimanov B. I., “Gamiltonovoct uravnenii Penleve i metod izomonodromnykh deformatsii”, Differ. uravneniya, 30:5 (1994), 791–796

[41] Culeimanov B. I., ““Kvantovaniya” vtorogo uravneniya Penleve i problema ekvivalentnosti ego $L,A$ par”, Teor. i mat. fiz., 156:3 (2008), 364–378

[42] Suleimanov B. I., ““Kvantovaniya” vysshikh gamiltonovykh analogov uravnenii Penleve I i II s dvumya stepenyami svobody”, Funkts. anal. i ego pril., 48:3 (2014), 52–62

[43] Culeimanov B. I., “Kvantovanie nekotorykh avtonomnykh reduktsii uravnenii Penleve i staraya kvantovaya teoriya”, Tezisy mezhdunar. konf., posvyaschennoi pamyati I. G. Petrovskogo, M., 2011, 356–357

[44] Suleimanov B. I., ““Kvantovaya” linearizatsiya uravnenii Penleve kak komponenta ikh $L-A$ par”, Ufimsk. mat. zh., 4:2 (2012), 127–135

[45] Suleimanov B. I., “Kvantovye aspekty integriruemosti tretego uravneniya Penleve i resheniya vremennogo uravneniya Shredingera s potentsialom Morsa”, Ufimsk. mat. zh., 8:3 (2016), 141–159

[46] Suleimanov B. I., “Vtoroe uravnenie Penleve v odnoi zadache o nelineinykh effektakh vblizi kaustiki”, Zap. nauch. semin. LOMI, 187, 1991, 110–128