How to find eigenfunctions for the Fourier transform.

Question

I was searching for information on wikipedia, when I've came across a differential equation to find eigenfunctions for the Fourier Transform: $$ \left [U\left(\frac{1}{2 \pi} \frac{d}{dx} \right) + U(x) \right ] \psi = 0 $$ Where $U(x)$, is a function which can be expressed as a Taylor series. But, my question is, how would you even arrive to this? I've tried to start from the definition, $$ \mathcal F [\psi] =\lambda \psi $$ but I couldn't do much with it. So, how would I arrive to the differential equation, from this starting point?