View source for Floquet exponents

{{TEX|done}}
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

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

where $u(t)$ is a vector function on $\mathbf R$ with values in a finite-dimensional complex [[Vector space|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|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

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

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 [[#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 $A(t)$ (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]] $-\Delta u+qu=0$ 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

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

and define the differential form $A=\sum A_jdx_j$. Assume that the electric potential $V$ and the [[Magnetic field|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====
<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>