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How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it?

The Mathieu operator is an ordinary periodic differential operator of the form

$$\frac{d^2y}{dx^2} + (a-2q \cos{2x})y =0.$$

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    It's pretty much the canonical example of Floquet's theory of differential equations with periodic coefficients. You might want to look [here](http://dlmf.nist.gov/28) for an overview and links to appropriate books/papers. (Sections 28.32 & 28.33 might be of particular interest.)2012-02-10

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