[fxj]

Yes the dirac comb for example. Actually there are infinitely many.

https://en.wikipedia.org/wiki/Dirac_comb

and for other:

http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/...

[abetusk]

A question I hadn't even thought to ask, thanks.

So, basically, the eigenfunctions of the Fourier transform are Hermite polynomials times a Gaussian [0] [1].

[0] https://math.stackexchange.com/questions/728670/functions-th...

[1] https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_fu...

[perihelions]

As well as the linear combinations (including infinite sums!) of Hermite functions with the same eigenvalue under the Fourier transform. (Those eigenvalues are infinitely degenerate). You could express sech(x) as such a sum.