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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?

Álvaro Rodrigo
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    It's not clear what you mean by "derive", since the word "derive" doesn't seem to be applicable to the approach taken here. It seems like what you're looking for is a motivation for this approach. In other words, the basic question is "how would you, as a mathematician who does not yet know the answer to the question, have thought of this approach?" Am I understanding the point of your question correctly? – Ben Grossmann Aug 13 '23 at 20:39
  • The Wikipedia page does not suggest to start from $\mathcal{F}[\psi]=\lambda\psi.$ That is just what it means to be an eigenfunction. I'm not sure that the equation containing $U$ can be derived somehow; it might as well just have been created from knowledge about how the Fourier transform works. – md2perpe Aug 13 '23 at 20:41
  • @BenGrossmann Yes, sorry, it is now edited. With "how to derive", I mean "how to arrive to". – Álvaro Rodrigo Aug 13 '23 at 20:59
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    Does this answer your question? What Eigenfunctions/distributions of the Fourier Transform exist? – Derivative Aug 14 '23 at 13:39

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