Floquet exponents

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].

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