# Is there a generalization of Floquet theory to elliptic functions?

Hi,

Consider a system of linear differential equations $${d f \over dz} = A(z) f,$$ where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, Floquet theorem applies.

I am curious to know if there exists a generalization of Floquet theorem to the case, where $z \in \mathbb{C}$ and $A(z)$ is a doubly-periodic elliptic function of $z$.

Thanks,

Victor

• The Lamé equation dlmf.nist.gov/29 seems to be the canonical example analogous to the Mathieu equation. – J. M. isn't a mathematician Oct 12 '10 at 6:31
• I don't know if it is relevant to your purpose, but the Wiki page says that the Floquet theorem generalizes to the Bloch theorem in higher dimensions. – timur Nov 1 '11 at 0:17

Another, far reaching, aspect of Floquet theory in differential equations $$a_0(z)f^{(n)}(z)+\cdots+a_nf(z)=0$$ with holomorphic coefficients is Fuchs theory of monodromy, where the leading coefficient $a_0$ has a zeros at $z_0$. You cannot solve a Cauchy problem at $z_0$, but you can solve it in a pointed disk $D\setminus z_0$. When you follow a circle around $z_0$, the coefficients look periodic.