Mathieu equation

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)

Comments

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)