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The following ordinary differential equation with real coefficients:
 
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>
+
$$
 +
 
 +
\frac{d  ^ {2} u }{dz  ^ {2} }
 +
+ ( a + b  \cos  2z) u  = 0,\ \
 +
z \in \mathbf R .
 +
$$
  
 
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]].
 
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]].
Line 7: Line 24:
 
A [[Fundamental system of solutions|fundamental system of solutions]] of the Mathieu equation has the form
 
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>
+
$$ \tag{* }
 +
u _ {1} ( z)  = e ^ {\alpha z } \phi ( z),\ \
 +
u _ {2} ( z)  = u _ {1} (- z) ,
 +
$$
  
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]]]).
+
for $  \alpha \neq ni $,  
 +
$  n $
 +
an integer, where $  \phi ( z) $
 +
is a $  \pi $-
 +
periodic function and the [[Lyapunov characteristic exponent|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|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 [[#References|[2]]], [[#References|[4]]]).
  
 
The Mathieu equation is known also in a different form (see [[#References|[3]]]).
 
The Mathieu equation is known also in a different form (see [[#References|[3]]]).
Line 15: Line 50:
 
====References====
 
====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>
 
<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====
 
====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
 
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>
+
$$
 +
( M _ {A , \alpha , \nu }  g ) ( n)  = \
 +
g ( n + 1 ) + 2 A  \cos ( 2 \pi n \alpha -
 +
\nu ) g ( n) + g ( n - 1 ) ,
 +
$$
  
<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>
+
$$
 +
A , \alpha , \nu  \in  \mathbf R ,
 +
$$
  
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
+
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
  
<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>
+
$$
 +
\mathop{\rm Spec} ( A , \alpha )  = \cup _  \nu
 +
\mathop{\rm Spec} ( A , \alpha , \nu ) .
 +
$$
  
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]]].
+
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 [[#References|[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. [[#References|[a2]]]. A selection of noteworthy papers on these matters as well as results for the continuous analogues is [[#References|[a3]]]–[[#References|[a5]]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 07:59, 6 June 2020


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)
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