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.



Spectrum of Mathieu equation
A good reference is Whittaker Watson, vol. 2.


