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Exponents arising in the study of solutions of a [[Linear ordinary differential equation|linear ordinary differential equation]] invariant with respect to a discrete [[Abelian group|Abelian group]] (cf. also [[Floquet theory|Floquet theory]]). The simplest example is a periodic ordinary differential equation
 
Exponents arising in the study of solutions of a [[Linear ordinary differential equation|linear ordinary differential equation]] invariant with respect to a discrete [[Abelian group|Abelian group]] (cf. also [[Floquet theory|Floquet theory]]). The simplest example is a periodic ordinary differential equation
  
<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/f/f120/f120070/f1200701.png" /></td> </tr></table>
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$$\frac{du}{dt}=A(t)u,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200702.png" /> is a vector function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200703.png" /> with values in a finite-dimensional complex [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200705.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200706.png" />-periodic function with values in the space of linear operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200707.png" />. The space of solutions of this equation is finite-dimensional and invariant with respect to the action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200708.png" /> of the integer group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f1200709.png" /> by shifts
+
where $u(t)$ is a vector function on $\mathbf R$ with values in a finite-dimensional complex [[Vector space|vector space]] $H$ and $A(t)$ is an $\omega$-periodic function with values in the space of linear operators in $H$. The space of solutions of this equation is finite-dimensional and invariant with respect to the action $T$ of the integer group $\mathbf Z$ by shifts
  
<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/f/f120/f120070/f12007010.png" /></td> </tr></table>
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$$(T(n)f)(x)=f(x+\omega n),\quad x\in\mathbf R,n\in\mathbf Z.$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007011.png" /> is the [[Monodromy operator|monodromy operator]]. One can expand any solution into eigenvectors and generalized eigenvectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007012.png" /> (cf. also [[Eigen vector|Eigen vector]]). This amounts to expanding the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007013.png" /> on the solution space into irreducible and primary representations (cf. also [[Representation of a group|Representation of a group]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007014.png" /> is an [[Eigen value|eigen value]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007016.png" /> is the corresponding eigenvector, then
+
Here, $M=T(1)$ is the [[Monodromy operator|monodromy operator]]. One can expand any solution into eigenvectors and generalized eigenvectors of $M$ (cf. also [[Eigen vector|Eigen vector]]). This amounts to expanding the action of $\mathbf Z$ on the solution space into irreducible and primary representations (cf. also [[Representation of a group|Representation of a group]]). If $\zeta$ is an [[Eigen value|eigen value]] of $M$ and $u(x)$ is the corresponding eigenvector, then
  
<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/f/f120/f120070/f12007017.png" /></td> </tr></table>
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$$u(x+n\omega)=\zeta^nu(x).$$
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007018.png" /> is the Floquet multiplier of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007019.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007022.png" /> is the Floquet exponent. The solution (called the Floquet solution, or the Bloch solution in physics) can be represented as
+
The number $\zeta$ is the Floquet multiplier of $u$. Since $\zeta\neq0$, $\zeta=\exp\omega d$, where $d$ is the Floquet exponent. The solution (called the Floquet solution, or the Bloch solution in physics) can be represented as
  
<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/f/f120/f120070/f12007023.png" /></td> </tr></table>
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$$u(x)=e^{dx}p(x),$$
  
with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007024.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007025.png" />. The generalized eigenvectors can be written as
+
with an $\omega$-periodic function $p(x)$. The generalized eigenvectors can be written as
  
<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/f/f120/f120070/f12007026.png" /></td> </tr></table>
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$$u(x)=e^{dx}\left(\sum_{j=0}^mx^jp_j(x)\right),$$
  
with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007027.png" />-periodic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007028.png" />. Floquet solutions play a major role in any considerations involving periodic ordinary differential equations, similar to exponential-polynomial solutions in the constant-coefficient case. This approach to periodic ordinary differential equations was developed by G. Floquet [[#References|[a4]]]. One can find detailed description and applications of this theory in many places, for instance in [[#References|[a3]]] and [[#References|[a7]]].
+
with an $\omega$-periodic $p_j(x)$. Floquet solutions play a major role in any considerations involving periodic ordinary differential equations, similar to exponential-polynomial solutions in the constant-coefficient case. This approach to periodic ordinary differential equations was developed by G. Floquet [[#References|[a4]]]. One can find detailed description and applications of this theory in many places, for instance in [[#References|[a3]]] and [[#References|[a7]]].
  
The Floquet theory can, to some extent, be carried over to the case of evolution equations in infinite-dimensional spaces with bounded or unbounded operator coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007029.png" /> (for instance, for the time-periodic heat equation; cf. also [[Heat equation|Heat equation]]). One can find discussion of this matter in [[#References|[a5]]] and [[#References|[a2]]].
+
The Floquet theory can, to some extent, be carried over to the case of evolution equations in infinite-dimensional spaces with bounded or unbounded operator coefficient $A(t)$ (for instance, for the time-periodic heat equation; cf. also [[Heat equation|Heat equation]]). One can find discussion of this matter in [[#References|[a5]]] and [[#References|[a2]]].
  
Consider now the case of a partial differential equation periodic with respect to several variables. Among the most important examples arising in applications is the [[Schrödinger equation|Schrödinger equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007031.png" /> with a potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007032.png" /> that is periodic with respect to a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007034.png" /> [[#References|[a1]]]. A Floquet solution has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007035.png" />, where the Floquet exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007036.png" /> is a vector and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007037.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007038.png" />-periodic. One should note that in physics the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007039.png" /> is called the quasi-momentum [[#References|[a1]]]. Transfer of Floquet theory to the case of spatially periodic partial differential equations is possible, but non-trivial. For instance, one cannot use the monodromy operator (see [[#References|[a3]]] and [[#References|[a6]]] for the Schrödinger case and [[#References|[a5]]] for more general considerations).
+
Consider now the case of a partial differential equation periodic with respect to several variables. Among the most important examples arising in applications is the [[Schrödinger equation|Schrödinger equation]] $-\Delta u+qu=0$ in $\mathbf R^n$ with a potential $q(x)$ that is periodic with respect to a lattice $\Gamma$ in $\mathbf R^n$ [[#References|[a1]]]. A Floquet solution has the form $u(x)=e^{dx}p(x)$, where the Floquet exponent $d$ is a vector and the function $p(x)$ is $\Gamma$-periodic. One should note that in physics the vector $k=-id$ is called the quasi-momentum [[#References|[a1]]]. Transfer of Floquet theory to the case of spatially periodic partial differential equations is possible, but non-trivial. For instance, one cannot use the monodromy operator (see [[#References|[a3]]] and [[#References|[a6]]] for the Schrödinger case and [[#References|[a5]]] for more general considerations).
  
 
In some cases an equation can be periodic with respect to an Abelian group whose action is not just translation. Consider the magnetic Schrödinger operator
 
In some cases an equation can be periodic with respect to an Abelian group whose action is not just translation. Consider the magnetic Schrödinger operator
  
<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/f/f120/f120070/f12007040.png" /></td> </tr></table>
+
$$\sum_j\left(-i\frac{\partial}{\partial x_j}+A_j(x)\right)^2+V(x)$$
  
and define the differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007041.png" />. Assume that the electric potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007042.png" /> and the [[Magnetic field|magnetic field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007043.png" /> are periodic with respect to a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007044.png" />. This does not guarantee periodicity of the equation itself. However, if one combines shifts by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120070/f12007045.png" /> with appropriate phase shifts, one gets a discrete group (cf. also [[Discrete subgroup|Discrete subgroup]]) with respect to which the equation is invariant. This group is non-commutative in general [[#References|[a8]]], so the standard Floquet theory does not apply. However, under a rationality condition, the group is commutative and a version of magnetic Floquet theory is applicable [[#References|[a8]]].
+
and define the differential form $A=\sum A_jdx_j$. Assume that the electric potential $V$ and the [[Magnetic field|magnetic field]] $B=dA$ are periodic with respect to a lattice $\Gamma$. This does not guarantee periodicity of the equation itself. However, if one combines shifts by elements of $\Gamma$ with appropriate phase shifts, one gets a discrete group (cf. also [[Discrete subgroup|Discrete subgroup]]) with respect to which the equation is invariant. This group is non-commutative in general [[#References|[a8]]], so the standard Floquet theory does not apply. However, under a rationality condition, the group is commutative and a version of magnetic Floquet theory is applicable [[#References|[a8]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.W. Ashcroft,  N.D. Mermin,  "Solid State Physics" , Holt, Rinehart&amp;Winston  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ju.L. Daleckii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''43''' , Amer. Math. Soc.  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.S.P. Eastham,  "The spectral theory of periodic differential equations" , Scottish Acad. Press  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Floquet,  "Sur les equations differentielles lineaires a coefficients periodique"  ''Ann. Ecole Norm. Ser. 2'' , '''12'''  (1883)  pp. 47–89</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Kuchment,  "Floquet theory for partial differential equations" , Birkhäuser  (1993)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics: Analysis of operators" , '''IV''' , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.A. Yakubovich,  V.M. Starzhinskii,  "Linear differential equations with periodic coefficients" , '''1, 2''' , Halsted Press&amp;Wiley  (1975)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Zak,  "Magnetic translation group"  ''Phys. Rev.'' , '''134'''  (1964)  pp. A1602–A1611</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.W. Ashcroft,  N.D. Mermin,  "Solid State Physics" , Holt, Rinehart&amp;Winston  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ju.L. Daleckii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''43''' , Amer. Math. Soc.  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.S.P. Eastham,  "The spectral theory of periodic differential equations" , Scottish Acad. Press  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Floquet,  "Sur les equations differentielles lineaires a coefficients periodique"  ''Ann. Ecole Norm. Ser. 2'' , '''12'''  (1883)  pp. 47–89</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Kuchment,  "Floquet theory for partial differential equations" , Birkhäuser  (1993)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics: Analysis of operators" , '''IV''' , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.A. Yakubovich,  V.M. Starzhinskii,  "Linear differential equations with periodic coefficients" , '''1, 2''' , Halsted Press&amp;Wiley  (1975)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Zak,  "Magnetic translation group"  ''Phys. Rev.'' , '''134'''  (1964)  pp. A1602–A1611</TD></TR></table>

Latest revision as of 16:01, 19 August 2014

Exponents arising in the study of solutions of a linear ordinary differential equation invariant with respect to a discrete Abelian group (cf. also Floquet theory). The simplest example is a periodic ordinary differential equation

$$\frac{du}{dt}=A(t)u,$$

where $u(t)$ is a vector function on $\mathbf R$ with values in a finite-dimensional complex vector space $H$ and $A(t)$ is an $\omega$-periodic function with values in the space of linear operators in $H$. The space of solutions of this equation is finite-dimensional and invariant with respect to the action $T$ of the integer group $\mathbf Z$ by shifts

$$(T(n)f)(x)=f(x+\omega n),\quad x\in\mathbf R,n\in\mathbf Z.$$

Here, $M=T(1)$ is the monodromy operator. One can expand any solution into eigenvectors and generalized eigenvectors of $M$ (cf. also Eigen vector). This amounts to expanding the action of $\mathbf Z$ on the solution space into irreducible and primary representations (cf. also Representation of a group). If $\zeta$ is an eigen value of $M$ and $u(x)$ is the corresponding eigenvector, then

$$u(x+n\omega)=\zeta^nu(x).$$

The number $\zeta$ is the Floquet multiplier of $u$. Since $\zeta\neq0$, $\zeta=\exp\omega d$, where $d$ is the Floquet exponent. The solution (called the Floquet solution, or the Bloch solution in physics) can be represented as

$$u(x)=e^{dx}p(x),$$

with an $\omega$-periodic function $p(x)$. The generalized eigenvectors can be written as

$$u(x)=e^{dx}\left(\sum_{j=0}^mx^jp_j(x)\right),$$

with an $\omega$-periodic $p_j(x)$. Floquet solutions play a major role in any considerations involving periodic ordinary differential equations, similar to exponential-polynomial solutions in the constant-coefficient case. This approach to periodic ordinary differential equations was developed by G. Floquet [a4]. One can find detailed description and applications of this theory in many places, for instance in [a3] and [a7].

The Floquet theory can, to some extent, be carried over to the case of evolution equations in infinite-dimensional spaces with bounded or unbounded operator coefficient $A(t)$ (for instance, for the time-periodic heat equation; cf. also Heat equation). One can find discussion of this matter in [a5] and [a2].

Consider now the case of a partial differential equation periodic with respect to several variables. Among the most important examples arising in applications is the Schrödinger equation $-\Delta u+qu=0$ in $\mathbf R^n$ with a potential $q(x)$ that is periodic with respect to a lattice $\Gamma$ in $\mathbf R^n$ [a1]. A Floquet solution has the form $u(x)=e^{dx}p(x)$, where the Floquet exponent $d$ is a vector and the function $p(x)$ is $\Gamma$-periodic. One should note that in physics the vector $k=-id$ is called the quasi-momentum [a1]. Transfer of Floquet theory to the case of spatially periodic partial differential equations is possible, but non-trivial. For instance, one cannot use the monodromy operator (see [a3] and [a6] for the Schrödinger case and [a5] for more general considerations).

In some cases an equation can be periodic with respect to an Abelian group whose action is not just translation. Consider the magnetic Schrödinger operator

$$\sum_j\left(-i\frac{\partial}{\partial x_j}+A_j(x)\right)^2+V(x)$$

and define the differential form $A=\sum A_jdx_j$. Assume that the electric potential $V$ and the magnetic field $B=dA$ are periodic with respect to a lattice $\Gamma$. This does not guarantee periodicity of the equation itself. However, if one combines shifts by elements of $\Gamma$ with appropriate phase shifts, one gets a discrete group (cf. also Discrete subgroup) with respect to which the equation is invariant. This group is non-commutative in general [a8], so the standard Floquet theory does not apply. However, under a rationality condition, the group is commutative and a version of magnetic Floquet theory is applicable [a8].

References

[a1] N.W. Ashcroft, N.D. Mermin, "Solid State Physics" , Holt, Rinehart&Winston (1976)
[a2] Ju.L. Daleckii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Transl. Math. Monogr. , 43 , Amer. Math. Soc. (1974)
[a3] M.S.P. Eastham, "The spectral theory of periodic differential equations" , Scottish Acad. Press (1973)
[a4] G. Floquet, "Sur les equations differentielles lineaires a coefficients periodique" Ann. Ecole Norm. Ser. 2 , 12 (1883) pp. 47–89
[a5] P. Kuchment, "Floquet theory for partial differential equations" , Birkhäuser (1993)
[a6] M. Reed, B. Simon, "Methods of modern mathematical physics: Analysis of operators" , IV , Acad. Press (1978)
[a7] V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients" , 1, 2 , Halsted Press&Wiley (1975)
[a8] J. Zak, "Magnetic translation group" Phys. Rev. , 134 (1964) pp. A1602–A1611
How to Cite This Entry:
Floquet exponents. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Floquet_exponents&oldid=14228
This article was adapted from an original article by P. Kuchment (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article