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Entire functions of finite order as solutions to certain complex linear differential equations
Author:
N. Anghel
Journal:
Proc. Amer. Math. Soc. 140 (2012), 2319-2332
MSC (2010):
Primary 30D15, 34M05; Secondary 33C10, 34L40
Posted:
October 3, 2011
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Additional Information
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.
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Orders, J. Math. Phys.,  No. 5, 053509 (2010). MR 2666987 (2011f:31009)
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- R. B. Ash, Complex Variables, Academic Press, Inc., New York (1971). MR 0281883 (43:7597)
- [BS]
- F. A. Berezin, M. Shubin, The Schrödinger Equation, Kluwer, Dordrecht (1991). MR 1186643 (93i:81001)
- [D]
- T. Duc Tai, On the Simpleness of Zeros of Stokes Multipliers, J. Differential Equations,
 , No. 2, 351-366 (2006). MR 2214939 (2006k:34242)
- [E]
- S. M. Elzaidi, On Bank-Laine Sequences, Complex Variables Theory Appl.,
 , No. 3, 201-220 (1999). MR 1694317 (2000a:34170)
- [EGS1]
- A. Eremenko, A. Gabrielov, B. Shapiro, Zeros of Eigenfunctions of Some Anharmonic Oscillators, Ann. Inst. Fourier,
 , No. 2, 603-624 (2008). MR 2410384 (2009b:30007)
- [EGS2]
- A. Eremenko, A. Gabrielov, B. Shapiro, High Energy Eigenfunctions of One-Dimensional Schrödinger Operators with Polynomial Potentials, Comput. Methods Funct. Theory,
 , No. 1-2, 513-529 (2008). MR 2419492 (2009d:81100)
- [F]
- M. Frei, Über die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten, Comment. Math. Helv.,
 , 201-222 (1961). MR 0126008 (23:A3305)
- [G]
- M. I. Gil', Identities for Sums of Powers of Roots of Entire Functions, Compl. Var. Ellipt. Eqns.,
 , No. 1, 63-68 (2006). MR 2201257 (2007h:30030)
- [L]
- I. Laine, Nevanlinna Theory and Complex Differential Equations, W. de Gruyter, Berlin, New York (1993). MR 1207139 (94d:34008)
- [N]
- J. Nikolaus, Lineare Differentialgleichungen mit gegebener ganzer Lösung, Math. Z.,
 , 30-36 (1968). MR 0222367 (36:5419)
- [S]
- Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland Publ. Co., Amsterdam (1975). MR 0486867 (58:6561)
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- T. Stieltjes, Sur Certains Polynômes qui Verifient une Équation Differentielle Linéaire du Second Ordre et sur la Théorie de Fonctions de Lamé, Acta Math.,
 , 321-326 (1885). MR 1554669
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- A. V. Turbiner, Quasi-Exactly-Solvable Problems and
 Algebra, Commun. Math. Phys.,  , 467-474 (1988). MR 958807 (89j:58051)
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- A. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, Inst. Physics Publ., Bristol (1994). MR 1329549 (96i:81297)
- [VS]
- O. Vallée, M. Soares, Airy Functions and Applications to Physics, Imperial College Press, London, 2004. MR 2114198 (2006c:33006)
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- H. Wittich, Zur Theorie Linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A,
 , 1-18 (1966). MR 0197828 (33:5989)
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 , 125-133 (1985). MR 771035 (86i:12001)
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Additional Information
N. Anghel
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203
Email:
anghel@unt.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11055-4
PII:
S 0002-9939(2011)11055-4
Keywords:
Complex differential equations,
polynomial coefficients,
entire functions,
finite order,
zeros,
Stieltjes relations,
Gil’ sums,
Schrödinger operators
Received by editor(s):
September 29, 2010
Received by editor(s) in revised form:
February 4, 2011
Posted:
October 3, 2011
Communicated by:
Walter Van Assche
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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