Spectrum of Mathieu equation

Question

I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = f(2\pi),f'(0) = f'(2\pi)$. I know that this equation is called a Mathieu equation and it is well-studied in the literature. Especially, there are a lot of numerical algorithms out there that would do it. Despite, I could not find much about explicite or recursive ways to study the spectrum. Therefore, I was wondering whether anybody could give me a few references where I can find something about this and maybe somebody is also able to give me a short overview of what can be done explicitely.

Answer 1

A good reference is Whittaker Watson, vol. 2.

Answer 2

The Wikipedia article (http://en.wikipedia.org/wiki/Mathieu_function) is quite extensive, I believe, and in general there is NO way to explicitly calculate either eigenvalues or eigenfunctions of the Mathieu equation.

The Wikipedia article also explicitly states that the solutions generally cannot be expressed in terms of hypergeometric functions.

You can do perturbation theory around $q=\infty$ and recursively calculate asymptotic expansions of eigenfunctions and eigenvalues to any order (semiclassical analysis, basically). However, you seem not to be interested in this. (A reference would be a quite old Paper, unfortunately in german: J. Meixner, "Asymptotische Entwicklung der Eigenwerte und Eigenfunktionen der Differentialgleichungen der Sphäroid-Funktionen und der Mathieuschen Funktionen", Zeitschrift füt angewandte Mathematik und Mechanik, 1948)

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Edit:

What I didn't mention if you are interested in approximations: Apart from the semiclassical expansion, the operator family $$H(q) = - \frac{d^2}{d x^2} - q \cos(x)$$ is a self-adjoint family of unbounded operators, where each operator has the same domain (the Sobolev space $H^2(S^1)$ on the circle). By general perturbation theory for these operators (the standard reference is Kato, "Perturbation Theory for Linear Operators"), the eigenvalues are holomorphic in some domain including the real axis, and the eigenspaces are finite-dimensional as well and parametrized by the parameter $q$ in a holomorphic way.

I believe that you can (theoretically) calculate all coefficients of the corresponding Taylor series at $q=0$, both of eigenvalues and eigenfunctions.

As opposed to the other case above, this expansion will actually converge in a neighborhood near zero (the semiclassical expansion is only asymptotic). If I recall correctly, the eigenvalues $a_n(q)$ and $b_n(q)$ are holomorphic on some Riemann surface covering $\mathbb{C}$ which sadly has a very complicated branch structure (there are no real branch points, however, as no splitting of eigenvalues occurs on the real axis in the self-adjoint case). The problem for your purpose is that these branch points will limit the radius of convergence of the Taylor series at zero. If you have explicit values of $q$ in mind, it may or may not be enough. The locus of these branchpoints should probably be known, but I do not have a reference.