I would be very thankful for any hints concerning the finding the eigenvalues and eigenfunctions $f(x)$ of the following operator defined on the segment $[0,2\pi]$: $$ -\frac{d^2}{dx^2}+V\cos mx, $$ $f(x)$ being infinitely differentiable on the whole segment $[0,2\pi]$ and being subject to the boundary conditions $$ \forall n\ge0:\quad f^{(n)}(2\pi)=f^{(n)}(0). $$ $V$ is a real constant, $m$ is a positive integer number.
Eigenvalues and eigenfunctions of a seemingly simple operator.
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ordinary-differential-equations
eigenvalues-eigenvectors
eigenfunctions
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0I'm wondering if a Fourier series (complex form?) approach is ideal here. Trigonometric potentials are a bit of a pain to work with. – 2017-02-10
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0I tried it, but don't see how to terminate the resulting series. – 2017-02-10
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0I've found out that for $m=2$ the equation is known as Mathieu's differential equation. If anybody knows the name for its generalization for arbitrary integer $m$ please let me know. – 2017-02-10
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0It doesn't have to terminate. But did you manage to find conditions for eigenvalues? – 2017-02-10
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0No, I did not. Do you have some idea? – 2017-02-15