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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 $z$.



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The Lamé equation seems to be the canonical example analogous to the Mathieu equation. – J. M. 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

After some searching around, it looks to me that it was Picard who thought of generalizing Floquet theory to linear ODEs with doubly-periodic coefficients (though according to this paper, Floquet himself worked a bit on this generalization, deriving his theory for ODEs with singly periodic coefficients from the more general case).

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Hi JM -- Thanks for the paper. Very useful. I definitely should improve my google-searching skills, as I could not find anything sensible. Anyway, I am sort of amazed that such a natural question is not discussed in detail in any but a very few papers. When this question first came up in my (physics) research, I was hoping to quickly find a book or review on it. All the best – Victor Galitski Oct 12 '10 at 13:49

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.

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