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How can I find Eigenvalues of the differential operator $-\frac{d^2}{dt^2} + \mathrm{sin} =\Delta + \mathrm{sin}$, defined on $M=S^1$, the 1-Sphere?

It seems that the underlying ODE is pretty much unsolvable explicitly, but after all I am only interested in the Eigenvalues, not the Eigenfunctions itself, so there might be some other way?

I am also interested in functions $v \in C^\infty(S^1)$ such that the EV of $\Delta + v$ can be computed easily, but I was not able to find any nonconstant ones yet.

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Looks like you basically have periodic Mathieu functions: en.wikipedia.org/wiki/Mathieu_function I expect that the spectrum cannot be obtained in closed form but much is known about it. –  Noam D. Elkies Sep 18 '11 at 15:28
    
What do you mean by "finding?" Exact values or numerical approximation, or good estimates? –  András Bátkai Sep 19 '11 at 13:03
    
The hint with Mathieu Functions was pretty much what I needed! Thank you very much. –  Matthias Ludewig Sep 22 '11 at 19:13

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