Geodesic
Method of re-quantization and its application to the construction of asymptotics for solutions of non-Fuchsian-type equations with holomorphic coefficients
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 4, Tome 173 (2019), pp. 72-85.

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In this paper, we apply methods of resurgent analysis (including the method of repeated quantization) to the construction of asymptotics for solutions of linear ordinary differential equations with holomorphic coefficients. We provide a classification of various types of asymptotics depending on the principal symbol of the differential operator. Using the method of repeated quantization, we construct asymptotics for solutions of an ordinary differential equation with holomorphic coefficients in a neighborhood of infinity.
Keywords: Fuchsian linear differential equation, irregular singular point, asymptotics, resurgent function, principal symbol of a differential operator
Mots-clés : Laplace–Borel transform, method of re-quantization. 
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     title = {Method of re-quantization and its application to the construction of asymptotics for solutions of {non-Fuchsian-type} equations with holomorphic coefficients},
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M. V. Korovina; V. Yu. Smirnov. Method of re-quantization and its application to the construction of asymptotics for solutions of non-Fuchsian-type equations with holomorphic coefficients. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 4, Tome 173 (2019), pp. 72-85. http://geodesic.mathdoc.fr/item/INTO_2019_173_a6/

[1] Arnold V. I., Ilyushenko Yu. S., “Obyknovennye differentsialnye uravneniya”, Dinamicheskie sistemy—1, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravl., 1, VINITI, M., 1985, 7–140

[2] Kats D. S., “Vychislenie asimptotik reshenii uravnenii s polinomialnymi vyrozhdeniyami koeffitsientov”, Differ. uravn., 51:12 (2015), 1612–1617 | DOI | Zbl

[3] Kats D. S., “Koeffitsienty ryadov v asimptoticheskikh razlozheniyakh reshenii uravnenii s vyrozhdeniyami”, Int. J. Open Inform. Tech., 4:9 (2016), 1–7

[4] Koddington E. A., Levinson N., Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1958

[5] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v konicheskikh oblastyakh”, Dokl. AN SSSR., 153:1 (1963), 27–29 | Zbl

[6] Korovina M. V., “Suschestvovanie resurgentnogo resheniya dlya uravnenii s vyrozhdeniem vysshikh poryadkov”, Differ. uravn., 47:3 (2011), 349–357 | MR

[7] Korovina M. V., “Asimptotiki reshenii uravnenii s vysshimi vyrozhdeniyami”, Dokl. RAN., 437:3 (2011), 302–304 | Zbl

[8] Korovina M. V., “Metod povtornogo kvantovaniya i ego primeneniya k postroeniyu asimptotik reshenii uravnenii s vyrozhdeniyami”, Differ. uravn., 52:1 (2016), 60–77 | DOI | Zbl

[9] Korovina M. V., Shatalov V. E., “Differentsialnye uravneniya s vyrozhdeniem i resurgentnyi analiz”, Differ. uravn., 46:9 (2010), 1259–1277 | MR | Zbl

[10] Korovina M. V., Shatalov V. E., “Differentsialnye uravneniya s vyrozhdeniem”, Dokl. RAN., 437:1 (2011), 16–19 | MR | Zbl

[11] Olver F., Asimptotika i spetsialnye funktsii, Nauka, M., 1990

[12] Sternin B. Yu., Shatalov V. E., “Differentsialnye uravneniya v prostranstvakh s asimptotikami na mnogoobraziyakh s osobennostyami tipa klyuva”, Differ. uravn., 38:12 (2002), 1664–1672, M. | MR | Zbl

[13] Chezari L., Asimptoticheskoe povedenie i ustoichivost reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1964

[14] Ecalle J., “Cinq applications des fonctions résurgentes”, Prepub. Math. d'Orsay, 84T62 (1984), 110

[15] Korovina M. V., Smirnov V. Yu., “Construction of asymptotics of solutions of differential equations with cusp-type degeneration in the coefficients in the case of multiple roots of the highest-order symbol”, Differ. Eqs., 54:1 (2018), 28–37 | DOI | MR | Zbl

[16] Korovina M. V., Smirnov V. Yu., “Linear differential equations with holomorphic coefficients on manifold with cuspidal singularities and Laplace's equation”, Mech. Mat. Sci. Eng., 19 (2018), 1–9

[17] Korovina M. V., Smirnov I. N., Smirnov V., “On a problem arising in application of the re-quantization method to construct asymptotics of solutions to linear differential equations with holomorphic coefficients at infinity”, Math. Comput. Appl., 24:1 (2019), 16 | MR

[18] Poincaré H., “Sur les integrales irregulieres des equations lineaires”, Acta Math., 8 (1886), 295–344 | DOI | MR

[19] Schulze B. W., Sternin B., Shatalov V., Asymptotic Solutions to Differential Equations on Manifolds with Cusps, preprint MPI 96-89, Max-Planck-Institut für Mathematik, Bonn, 1996 | MR

[20] Schulze B. W., Sternin B., Shatalov V., “An Operator Algebra on Manifolds with Cusp-Type Singularities”, Ann. Glob. Anal. Geom., 16:2 (1998), 101–140 | DOI | MR | Zbl

[21] Sternberg W., Über die asymptotische Integration von Differencialgleichungen, Spriger, Berlin, 1920 | MR

[22] Sternin B., Shatalov V., Borel–Laplace Transform and Asymptotic Theory. Introduction to Resurgent Analysis, CRC Press, 1996 | MR | Zbl

[23] Thomé L. W., “Zur Theorie der linearen Differentialgleichungen”, J. Reine Angew. Math., 96:3 (1884), 185–281 | MR