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.