Some of the "basic" Fourier eigenfuctions encountered in real and complex analysis are
$$ e^{-\pi x^2}, \quad \frac{1}{\cosh{\pi x}} $$
What are some other (smooth) functions fixed under the Fourier transform which have different decay rates than the above two functions? (1)
For $f\in{\mathcal{S}(\mathbb{R})}$, define
$$\widehat{f}(\xi) = \int_{\mathbb{R}} f(x) e^{-2 \pi i x \xi} dx $$
Some background: It is possible to generate additional Fourier eigenfuctions by means of various iterative processes. A classical example are the Hermite functions, which may be given by
$$h_k(x) = (-1)^k e^{x^2/2} \frac{d^k}{dx^k} \left( e^{-x^2} \right) $$
$h_0(\sqrt{2 \pi} x) = e^{-\pi x^2}$ and it turns out that $H_k(x) = h_k(\sqrt{2 \pi} x) $ are Fourier eigenfuctions with eigenvalue $(-i)^k$. So, if $\widehat{f} = f$, then it is possible to write
$$ f \sim \sum_m a_{4m}H_{4m} $$
It appears difficult to find the/any correct combination of Hermite functions which create a +1 function satisfying (1).
Alternatively, if we set $ \gamma(x) $ equal to $e^{-\pi x^2} $ or $ \mathrm{sech}\pi x$ from above, then for $a > 1$, the functions $\gamma(x)$, $a\gamma(ax) + \gamma(x/a) $, and $a\gamma(ax) + \gamma(x/a) - (1+a)\gamma(x) $ are +1 eigenfunctions due directly to the dilation property $ f(\delta x) \longrightarrow 1/ \delta \widehat{f}(\xi / \delta) $. The following plot shows these three functions with $ \gamma(x) = \mathrm{sech} \pi x $ and $a=1.5$.
Although $a \gamma(ax) + \gamma(x/a)$ is another eigenfunction, it's not fundamentally different from $\gamma(x)$. One "problem" (at least in my case) with $ a\gamma(ax) + \gamma(x/a) - (1+a)\gamma(x)$ (in green) is that it oscillates. The higher order Hermite functions also oscillate before decaying rapidly. $\mathrm{sech} \pi x$ decays like $e^{-|x|}$ while the Hermite functions, which take the form $e^{-x^2/2} P(x)$ (P a ploynomial) decay like $e^{-|x|^2}$. I'm wondering if there are other +1 (self-dual) functions which decay at different rates. In a sense, I'm trying to reverse an iteration or generating relation, and one possible avenue may be to consider functions of the form
$$ \int_1^\infty (a\gamma(ax) + \gamma(x/a) - (1+a)\gamma(x)) d \sigma (a) $$ where $\sigma$ is a (positive?) measure on $(1,\infty)$. However, it appears that pinning down any particular such measures which make the integral converge is difficult.