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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 for an overview and links to appropriate books/papers. (Sections 28.32 & 28.33 might be of particular interest.) –  J. M. Feb 10 '12 at 6:31
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The Mathieu functions naturally arise when one solves problems for Laplace/Hemlholtz/wave equations in elliptic coordinates using the method of separation of variables. For example, Mathieu functions can be used to describe the natural modes of elliptic plates. Just as Bessel functions are basic separable solutions for circles and spheres, Mathieu functions are basic separable solutions for elliptic cylinders.

Their importance largely depends on whether you are interested in elliptic geometries. Generally speaking, until recently, people tried to avoid elliptic geometries because Mathieu functions are awkward to compute and many mathematical libraries did not implement them. Nowadays this is not an issue; both Maple and Mathematical have good numerical implementations of Mathieu functions. The best book on the theory of Mathieu functions was written by McLachlan "Theory and application of Mathieu functions".

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For better understanding of Mathieu functions I recommend a classical book by M. Abramovitz and I. Stegun and Mathieu Functions Toolbox implemented in Scilab.

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This subject, especially 1D case, is based on Floquet Theory. It's important to study Mathieu operator's spectral properties; e.g asymptotics of its spectral gaps subject to some specific boundary conditions.

These spectral gaps is related with the notion of tunneling in momentum space. "Of particular interest are the gaps, which are the size of various forbidden regions of the spectrum in the one electron theory of solids or alternatively regions of instability in the theory of parametric resonance in classical mechanics." J. Avron- B. Simon.

The books of S. Winkler and W. Magnus, "The coexistence problem for Hill's equation" and M. S. P. Eastham, "The spectral theory of periodic dierential operators" are the basics for Floquet Theory and in particular, for the Mathieu operator.

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The book by Winkler & Magnus is called simply "Hill's equation". –  Aleksey Pichugin Mar 11 '12 at 15:49
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