Mathieu equation

The following ordinary differential equation with real coefficients:

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

(*)

for , an integer, where is a -periodic function and the Lyapunov characteristic exponent is either real or purely imaginary. For one of the solutions grows unboundedly, whereas the other tends to zero as (instability zones in the plane of the parameters ); for 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 -periodic or -periodic (the latter is called a Mathieu function, cf. Mathieu functions), while the second is obtained from the first through multiplication by . The instability zones have the form of curvilinear triangles with vertices at the points , , (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

is important. If is rational this is a periodic operator, otherwise it is almost periodic. Let be the spectrum of on and let

The spectrum as a function of 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 is a Cantor set for all irrational , , ; another conjecture states that the Lebesgue measure of is zero for all irrational . For some detailed results on these spectra for rational 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)