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Borrego-Morell, Jorge A.

An asymptotic expansion of eigenpolynomials for a class of linear differential operators. (English) Zbl 1528.34069

Stud. Appl. Math. 151, No. 3, 923-956 (2023).
Summary: Consider an \(M\)-th order linear differential operator, \(M \geq 2\), \[ \mathcal{L}^{(M)} = \sum_{k=0}^M \rho_k(z)\frac{d^k}{dz^k}, \] where \(\rho_M\) is a monic complex polynomial such that \(\deg[\rho_M] = M\) and \((\rho_k)_{k=0}^{M-1}\) are complex polynomials such that \(\deg[\rho_k] \leq k\), \(0 \leq k \leq M-1\). It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure \(\mu\). We obtain an asymptotic expansion of the eigenpolynomials of \(\mathcal{L}^{(M)}\) in compact subsets out of the support of \(\mu\). In particular, we solve a conjecture posed in [G. Másson and B. Shapiro, Exp. Math. 10, No. 4, 609–618 (2001; Zbl 1008.33008)].
© 2023 Wiley Periodicals LLC.

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47E05 General theory of ordinary differential operators

Keywords:

asymptotic expansions; exactly solvable operators; ordinary differential equations; polynomials eigenfunctions; WKB solution

Citations:

Zbl 1008.33008
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Full Text: DOI arXiv

References:

[1] TurbinerA.Lie algebras and polynomials in one variable. J Phys A Math Gen. 1992;25(18):L1087. · Zbl 0764.34005
[2] EverittWN, KwonKH, LittlejohnLL, WellmanR.Orthogonal polynomial solutions of linear ordinary differential equations. J Comput Appl Math. 2001;133(1-2):85‐109. · Zbl 0993.33004
[3] HorozovE, ShapiroB, TaterM. In search of higher Bochner theorem. J. Approx. Theory. 2023; forthcoming.
[4] ShefferIM.On the properties of polynomials satisfying a linear differential equation: Part I. Trans Am Math Soc. 1933;35(1):184‐214. · Zbl 0006.16301
[5] MassonG, ShapiroB.On polynomial eigenfunctions of a hypergeometric type operator. Exper Math. 2001;10:609‐618. · Zbl 1008.33008
[6] BergkvistT, RullgårdH.On polynomial eigenfunctions for a class of differential operators. Math Res Lett. 2002;9:153‐171. · Zbl 1016.34083
[7] BergkvistT.On asymptotics of polynomial eigenfunctions for exactly solvable differential operators. J Approx Theory. 2007;149:151‐187. · Zbl 1155.34045
[8] BergkvistT, RullgårdH, ShapiroB.On Bochner-Krall orthogonal polynomial systems. Math Scand. 2004;94(1):148‐154. · Zbl 1072.33004
[9] ComtetL. Advanced Combinatorics: The Art of Finite and Infinite Expansions. D Reidel Publishing Company; 1974. · Zbl 0283.05001
[10] WentzelG.Eine verallgemeinerung der quantenbedingungen für die zwecke der wellenmechanik. Z Phys. 1926;38(6‐7):518‐529. · JFM 52.0969.03
[11] KramersHA.Wellenmechanik und halbzahlige quantisierung. Z Phys. 1926;39(10‐11):828‐840. · JFM 52.0969.04
[12] BrillouinL.La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives. CR. 1926;183:24‐26. · JFM 52.0967.05
[13] LiouvilleJ.Sur le développement des fonctions et séries. J Math Pures Appl. 1837;1:16‐37.
[14] GreenG.On the motion of waves in a variable canal of small depth and width. Trans Camb Phil Soc. 1837;6:457‐462.
[15] DingleRB. Asymptotic Expansions: Their Derivation and Interpretation. Academic Press; 1973. · Zbl 0279.41030
[16] WasowW. Linear Turning Point Theory, Volume 54 of Applied Mathematical Sciences. Springer‐Verlag; 1985. · Zbl 0558.34049
[17] OlverFWJ. Asymptotics and Special Functions. A K Peters, Ltd.; 1974.
[18] BenderCM, WuTT.Anharmonic oscillator. Phys Rev. 1969;184(5):1231‐1260.
[19] VorosA.The return of the quartic oscillator. The complex WKB method. Inst H Poincaré Sect A (NS). 1983;39(3):211‐338. · Zbl 0526.34046
[20] AokiT, KawaiT, TakeiY. The Bender-Wu analysis and the Voros theory. In: Kashiwara M, Miwa T, eds. ICM‐90 Satellite Conference Proceedings. Springer; 1991:1‐29. · Zbl 0782.35060
[21] AokiT, KawaiT, TakeiY.The Bender-Wu analysis and the Voros theory II. Adv Stud Pure Math. 2009;54:19‐34. · Zbl 1175.34112
[22] BodineS, SchäfkeR.On the summability of formal solutions in Liouville-Green theory. J Dyn Control. 2002;8(3):371‐398. · Zbl 1017.34089
[23] DelabaereE, PhamF.Resurgent methods in semi‐classical asymptotics. Ann IHP Phys Théor. 1999;71(1):1‐94. · Zbl 0977.34053
[24] KawaiT, TakeiY. Algebraic Analysis of Singular Perturbation Theory. Vol. 227. American Mathematical Soc.; 2005. · Zbl 1100.34004
[25] NemesG.On the Borel summability of WKB solutions of certain Schrödinger‐type differential equations. J Approx Theory. 2021;265:105562. · Zbl 1472.34158
[26] AokiT, KawaiT, TakeiY. New turning points in the exact WKB analysis for higher-order ordinary differential equations. I. Méthodes resurgentes. In: Boutet de Monvel L, ed. Analyse algébrique des perturbations singuliéres I. Hermann, Paris; 1994:69‐84. · Zbl 0831.34058
[27] FedoryukMV. Asymptotic Analysis of Linear Ordinary Differential Equations. Springer-Verlag; 1993. · Zbl 0782.34001
[28] KawaiT, KoikeT, TakeiY.On the exact WKB analysis of higher order simple-pole type operators. Adv Math. 2011;228(1):63‐96. · Zbl 1231.34155
[29] MotekiT, TakeiY.Stokes geometry of higher order linear ordinary differential equations and middle convolution. Adv Math. 2017;310:327‐376. · Zbl 1365.34149
[30] TakeiY. WKB analysis and Stokes geometry of differential equations. In: Filipuk G, Haraoka Y, Michalik S, eds. Analytic, Algebraic and Geometric Aspects of Differential Equations. Birkhäuser, Basel; 2017:263‐304. · Zbl 1404.34110
[31] KatzN. Rigid Local Systems, Volume 139 of Annals of Mathematics Studies. Princeton University Press; 1996. · Zbl 0864.14013
[32] DettweilerM, ReiterS.Middle convolution of Fuchsian systems and the construction of rigid differential systems. J Algebra. 2007;318:1‐24. · Zbl 1183.34137
[33] OshimaT. Fractional Calculus of Weyl Algebra and Fuchsian Differential Equations, Volume 28 of MSJ Memoirs. Math. Soc. Japan; 2012. · Zbl 1269.34003
[34] HsiehP‐F, SibuyaY. Basic Theory of Ordinary Differential Equations. Springer; 1999. · Zbl 0924.34001
[35] SibuyaY.Sur réduction analytique d’un système d’équations différentielles ordinaires linéaires contenant un paramètre. J Fac Sci Univ Tokyo. 1958;7(5):527‐540. · Zbl 0081.08103
[36] HornRA, JohnsonCR. Matrix Analysis. 2nd ed. Cambridge Universtiy Press; 2013. · Zbl 1267.15001
[37] MarkushevichAI. Theory of Functions of a Complex Variable. Prentice‐Hall; 1965. Translated from the Russian by R. A. Silverman. · Zbl 0135.12002
[38] StokesGG.On the discontinuity of arbitrary constants which appear in divergent developments. Trans Camb Phil Soc. 1864;10:106‐128.
[39] StokesGG. Mathematical and Physical Papers. Vol. IV. Cambridge University Press; 1904. · JFM 35.0032.05
[40] AblowitzMJ, FokasAS. Complex Variables: Introduction and Applications. Cambridge University Press; 2003. · Zbl 1088.30001
[41] ParisRB, WoodAD.Stokes phenomenon demystified. Bull Inst Math Appl. 1995;31(1-2):21‐28. · Zbl 0827.34002
[42] HondaN, KawaiT, TakeiY. Virtual Turning Points. Vol. 4. Springer; 2015. · Zbl 1354.34003
[43] KwonK, LittlejohnL, YoonG. Bochner-Krall orthogonal polynomials. In: DunklC (ed.), IsmailM (ed.), WongR (ed.), eds. Special Functions. World Scientific; 2000:181‐193. · Zbl 1097.42504
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