Entire functions of finite order as solutions to certain complex linear differential equations
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Abstract:
When is an entire function of finite order a solution to a complex 2nd order homogeneous linear differential equation with polynomial coefficients? In this paper we will give two (equivalent) answers to this question. The starting point of both answers is the Hadamard product representation of a given entire function of finite order. While the first answer involves certain Stieltjes-like relations associated to the function, the second one requires the vanishing of all but finitely many suitable expressions constructed via the Gil’ sums of the zeros of the function. Applications of these results will also be given, most notably to the spectral theory of one-dimensional Schrödinger operators with polynomial potentials.References
- N. Anghel, Stieltjes-Calogero-Gil’ relations associated to entire functions of finite order, J. Math. Phys. 51 (2010), no. 5, 053509, 9. MR 2666987, DOI 10.1063/1.3421667
- Robert B. Ash, Complex variables, Academic Press, New York-London, 1971. MR 0281883
- F. A. Berezin and M. A. Shubin, The Schrödinger equation, Mathematics and its Applications (Soviet Series), vol. 66, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1983 Russian edition by Yu. Rajabov, D. A. Leĭtes and N. A. Sakharova and revised by Shubin; With contributions by G. L. Litvinov and Leĭtes. MR 1186643, DOI 10.1007/978-94-011-3154-4
- Trinh Duc Tai, On the simpleness of zeros of Stokes multipliers, J. Differential Equations 223 (2006), no. 2, 351–366. MR 2214939, DOI 10.1016/j.jde.2005.07.020
- S. M. Elzaidi, On Bank-Laine sequences, Complex Variables Theory Appl. 38 (1999), no. 3, 201–220. MR 1694317, DOI 10.1080/17476939908815165
- Alexandre Eremenko, Andrei Gabrielov, and Boris Shapiro, Zeros of eigenfunctions of some anharmonic oscillators, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 2, 603–624 (English, with English and French summaries). MR 2410384, DOI 10.5802/aif.2362
- Alexandre Eremenko, Andrei Gabrielov, and Boris Shapiro, High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial potentials, Comput. Methods Funct. Theory 8 (2008), no. 1-2, 513–529. MR 2419492, DOI 10.1007/BF03321702
- Margrit Frei, Über die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten, Comment. Math. Helv. 35 (1961), 201–222 (German). MR 126008, DOI 10.1007/BF02567016
- M. I. Gil’, Identities for sums of powers of roots of entire functions, Complex Var. Elliptic Equ. 51 (2006), no. 1, 63–68. MR 2201257, DOI 10.1080/02781070500327766
- Ilpo Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR 1207139, DOI 10.1515/9783110863147
- J. Nikolaus, Lineare Differentialgleichungen mit gegebener ganzer Lösung, Math. Z. 103 (1968), 30–36 (German). MR 222367, DOI 10.1007/BF01111283
- Yasutaka Sibuya, Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Mathematics Studies, Vol. 18, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0486867
- T. J. Stieltjes, Sur certains polynômes, Acta Math. 6 (1885), no. 1, 321–326 (French). Qui vérifient une équation différentielle linéaire du second ordre et sur la theorie des fonctions de Lamé. MR 1554669, DOI 10.1007/BF02400421
- A. V. Turbiner, Quasi-exactly-solvable problems and $\textrm {sl}(2)$ algebra, Comm. Math. Phys. 118 (1988), no. 3, 467–474. MR 958807, DOI 10.1007/BF01466727
- Alexander G. Ushveridze, Quasi-exactly solvable models in quantum mechanics, Institute of Physics Publishing, Bristol, 1994. MR 1329549
- Olivier Vallée and Manuel Soares, Airy functions and applications to physics, Imperial College Press, London; Distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. MR 2114198, DOI 10.1142/p345
- H. Wittich, Zur Theorie linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A I No. 379 (1966), 19 (German). MR 0197828
- Alden H. Wright, Finding all solutions to a system of polynomial equations, Math. Comp. 44 (1985), no. 169, 125–133. MR 771035, DOI 10.1090/S0025-5718-1985-0771035-4
Additional Information
- N. Anghel
- Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
- MR Author ID: 26280
- Email: anghel@unt.edu
- Received by editor(s): September 29, 2010
- Received by editor(s) in revised form: February 4, 2011
- Published electronically: October 3, 2011
- Communicated by: Walter Van Assche
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2319-2332
- MSC (2010): Primary 30D15, 34M05; Secondary 33C10, 34L40
- DOI: https://doi.org/10.1090/S0002-9939-2011-11055-4
- MathSciNet review: 2898695