View source for Mathieu equation

The following ordinary differential equation with real coefficients:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627501.png" /></td> </tr></table>

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

A [[Fundamental system of solutions|fundamental system of solutions]] of the Mathieu equation has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627504.png" /> an integer, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627505.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627506.png" />-periodic function and the [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627507.png" /> is either real or purely imaginary. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627508.png" /> one of the solutions grows unboundedly, whereas the other tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m0627509.png" /> (instability zones in the plane of the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275010.png" />); for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275012.png" />-periodic or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275013.png" />-periodic (the latter is called a Mathieu function, cf. [[Mathieu functions|Mathieu functions]]), while the second is obtained from the first through multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275014.png" />. The instability zones have the form of curvilinear triangles with vertices at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275017.png" /> (see [[#References|[2]]], [[#References|[4]]]).

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

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



====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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275018.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275019.png" /></td> </tr></table>

is important. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275020.png" /> is rational this is a periodic operator, otherwise it is almost periodic. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275021.png" /> be the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275023.png" /> and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275024.png" /></td> </tr></table>

The spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275025.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275026.png" /> gives a figure in the plane with remarkable combinatorial regularity and Cantor set like properties. It is known as Hofstadter's butterfly [[#References|[a1]]]. M. Kac conjectured (the Martini problem) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275027.png" /> is a Cantor set for all irrational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275030.png" />; another conjecture states that the Lebesgue measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275031.png" /> is zero for all irrational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275032.png" />. For some detailed results on these spectra for rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062750/m06275033.png" /> and a survey of this problem area cf. [[#References|[a2]]]. A selection of noteworthy papers on these matters as well as results for the continuous analogues is [[#References|[a3]]]–[[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Hofstadter,   "The energy levels of Bloch electrons in rational and irrational magnetic fields"  ''Phys. Rev.'' , '''B14'''  (1976)  pp. 2239–2249</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Bélissard,   B. Simon,   "Cantor spectrum for the almost Mathieu potential"  ''J. Funct. Anal.'' , '''48'''  (1982)  pp. 408–419</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Simon,   "Almost periodic Schrödinger operators, a review"  ''Adv. Appl. Math.'' , '''3'''  (1982)  pp. 463–490</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Meixner,   F.W. Schäfke,   "Mathieu functions and spheroidal functions and their mathematical foundations: further studies" , Springer  (1980)</TD></TR></table>