Eigenvalues and eigenfunctions of a seemingly simple operator.

Question

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.