# Mathieu equation

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The following ordinary differential equation with real coefficients:

$$\frac{d ^ {2} u }{dz ^ {2} } + ( a + b \cos 2z) u = 0,\ \ z \in \mathbf R .$$

It was introduced by E. Mathieu [1] in the investigation of the oscillations of an elliptic membrane; it is a particular case of a Hill equation.

A fundamental system of solutions of the Mathieu equation has the form

$$\tag{* } u _ {1} ( z) = e ^ {\alpha z } \phi ( z),\ \ u _ {2} ( z) = u _ {1} (- z) ,$$

for $\alpha \neq ni$, $n$ an integer, where $\phi ( z)$ is a $\pi$- periodic function and the Lyapunov characteristic exponent $\alpha$ is either real or purely imaginary. For $\mathop{\rm Im} \alpha = 0$ one of the solutions grows unboundedly, whereas the other tends to zero as $z \rightarrow + \infty$( instability zones in the plane of the parameters $a , b$); for $\mathop{\rm Re} \alpha = 0$ these solutions are both bounded (stability zones). On the boundary of these zones (the case excluded in (*)) one of the functions of the fundamental system of solutions is either $\pi$- periodic or $2 \pi$- periodic (the latter is called a Mathieu function, cf. Mathieu functions), while the second is obtained from the first through multiplication by $z$. The instability zones have the form of curvilinear triangles with vertices at the points $a = n ^ {2}$, $b = 0$, $n = 0, 1 ,\dots$( see [2], [4]).

The Mathieu equation is known also in a different form (see [3]).

#### References

 [1] E. Mathieu, "Course de physique mathématique" , Paris (1873) [2] M.J.O. Strett, "Lamésche-, Mathieusche- und verwandte Funktionen in Physik und Technik" , Springer (1932) [3] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971) [4] V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients and their applications" , 1–2 , Wiley (1975) (Translated from Russian)

The operator involved in the Mathieu equation is called a Mathieu operator. In various applications, especially in solid state theory, a discrete analogue, the discrete Mathieu operator, defined by

$$( M _ {A , \alpha , \nu } g ) ( n) = \ g ( n + 1 ) + 2 A \cos ( 2 \pi n \alpha - \nu ) g ( n) + g ( n - 1 ) ,$$

$$A , \alpha , \nu \in \mathbf R ,$$

is important. If $\alpha$ is rational this is a periodic operator, otherwise it is almost periodic. Let $\mathop{\rm Spec} ( A , \alpha , \nu )$ be the spectrum of $M _ {A , \alpha , \nu }$ on $l _ {2} ( \mathbf Z )$ and let

$$\mathop{\rm Spec} ( A , \alpha ) = \cup _ \nu \mathop{\rm Spec} ( A , \alpha , \nu ) .$$

The spectrum $\mathop{\rm Spec} ( 1 , \alpha )$ as a function of $\alpha$ gives a figure in the plane with remarkable combinatorial regularity and Cantor set like properties. It is known as Hofstadter's butterfly [a1]. M. Kac conjectured (the Martini problem) that $\mathop{\rm Spec} ( A , \alpha , \nu )$ is a Cantor set for all irrational $\alpha$, $A \neq 0$, $\nu \in \mathbf R$; another conjecture states that the Lebesgue measure of $\mathop{\rm Spec} ( 1 , \alpha )$ is zero for all irrational $\alpha$. For some detailed results on these spectra for rational $\alpha$ and a survey of this problem area cf. [a2]. A selection of noteworthy papers on these matters as well as results for the continuous analogues is [a3][a5].

#### References

 [a1] D. Hofstadter, "The energy levels of Bloch electrons in rational and irrational magnetic fields" Phys. Rev. , B14 (1976) pp. 2239–2249 [a2] P.M.M. van Mouché, "Sur les régions interdites du spectre de l'opérateur périodique et discret de Mathieu" , Math. Inst. Univ. Utrecht (1988) (Thesis) [a3] J. Bélissard, B. Simon, "Cantor spectrum for the almost Mathieu potential" J. Funct. Anal. , 48 (1982) pp. 408–419 [a4] J. Bélissard, R. Lima, D. Testarel, "Almost periodic Schrödinger operators" L. Streit (ed.) , Mathematics and Physics, lectures on recent results , 1 , World Sci. (1985) pp. 1–64 [a5] B. Simon, "Almost periodic Schrödinger operators, a review" Adv. Appl. Math. , 3 (1982) pp. 463–490 [a6] J. Meixner, F.W. Schäfke, "Mathieu functions and spheroidal functions and their mathematical foundations: further studies" , Springer (1980)
How to Cite This Entry:
Mathieu equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mathieu_equation&oldid=47790
This article was adapted from an original article by V.M. Starzhinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article