Algebraic solutions of the Lamé equation, revisited

Robert S. Maier

doi:10.1016/j.jde.2003.06.006

Algebraic solutions of the Lamé equation, revisited

Robert S. Maier

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S₄, then the degree parameter of the equation may differ by ± 1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.

Original language	English (US)
Pages (from-to)	16-34
Number of pages	19
Journal	Journal of Differential Equations
Volume	198
Issue number	1
DOIs	https://doi.org/10.1016/j.jde.2003.06.006
State	Published - Mar 20 2004

Keywords

Algebraic solution
Finite monodromy
Hypergeometric equation
Lamé equation
Projective monodromy group
Schwarz list

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2003.06.006

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title = "Algebraic solutions of the Lam{\'e} equation, revisited",

abstract = "A minor error in the necessary conditions for the algebraic form of the Lam{\'e} equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S4, then the degree parameter of the equation may differ by ± 1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lam{\'e} equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.",

keywords = "Algebraic solution, Finite monodromy, Hypergeometric equation, Lam{\'e} equation, Projective monodromy group, Schwarz list",

author = "Maier, {Robert S.}",

note = "Funding Information: ·Fax: +520-621-6892. E-mail address: [email protected]. URL: http://www.math.arizona.edu/~rsm. 1Partially supported by NSF grants PHY-9800979 and PHY-0099484.",

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doi = "10.1016/j.jde.2003.06.006",

language = "English (US)",

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AU - Maier, Robert S.

N1 - Funding Information: ·Fax: +520-621-6892. E-mail address: [email protected]. URL: http://www.math.arizona.edu/~rsm. 1Partially supported by NSF grants PHY-9800979 and PHY-0099484.

PY - 2004/3/20

Y1 - 2004/3/20

N2 - A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S4, then the degree parameter of the equation may differ by ± 1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.

AB - A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S4, then the degree parameter of the equation may differ by ± 1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.

KW - Algebraic solution

KW - Finite monodromy

KW - Hypergeometric equation

KW - Lamé equation

KW - Projective monodromy group

KW - Schwarz list

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JO - Journal of Differential Equations

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ER -

Algebraic solutions of the Lamé equation, revisited

Abstract

Keywords

ASJC Scopus subject areas

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