Are there obvious eigenfunctions to the linear operator $(1+\mathcal{F})$ or $(1-\mathcal{F})$, where $\mathcal{F}$ is a linear operator, specifically the Fourier transform, such that $$F(x) = \mathcal{F} f(x) \equiv \int_{-\infty}^{\infty} f(s) \exp(2\pi i\,s\, x ) ds$$
This isn't relevant to the question, but the product of the above two terms selects the odd part of functions, and thus isn't invertible because there is generally loss of information in discarding the even part of the function.
Since $f(s) = e^{-\alpha s^2}$ is an eigenfunction of $\mathcal F$ for some $\alpha>0$ then it's obviously also an eigenfunction of both $1+\mathcal F$ and $1-\mathcal F$.