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I'm trying to find the eigenvalues and eigenvectors of the operator ##\hat{O}=\frac{d^2}{d\phi^2}##

Where ##\phi## is the angular coordinate in polar coordinates.

Since we are dealing with polar coordinates, we also have the condition (on the eigenfunctions) that ##f(\phi)=f(\phi+2\pi)##.

This problem leads us to the differential equation ##\hat{O}f=qf## where q is the eigenvalue corresponding to the eigenfunction ##f##.

The solution that I am looking at then concludes that the eigenfunctions take the form:$$f_q(x)=Ae^{\pm \sqrt{q} \phi}$$

But I don't see why this supposedly the solution instead of:$$f_q(x)=Ae^{\sqrt{q} \phi}+Be^{-\sqrt{q} \phi}$$

The second solution is the one I came up with, but I was unable to solve for the "spectrum of the operator" or the allowed values of q. That's why I tried to find an alternative solution.

I hope you can see my confusion. I understand how to find the spectrum given the first result, but I'm still not convinced that the first result is correct. After all, the general solution to the differential equation is what I got for my result. Furthermore, I can use the periodicity condition to solve for ##B## in terms of ##A## but for some reason my end result is not periodic. Can someone please provide some insight into the problem? Thank you

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# Eigenstuff of Second Derivative

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