In the space of smooth functions of one variable is there a way to tell what are the eigenfunctions of the operator $\exp(\partial_x)$, i.e. what is the solution of the eigenvalue problem
$e^{\partial_x}f(x)=af(x)$
seeking for the eigenvectors. I am asking since I don't see the solution to this, whereas asking for the eigenfunctions of $\partial_x$, i.e. solving the equation $\partial_xf(x)=af(x)$ is easily done via $f(x)=be^{cx+d}$.
Can this be done easily or is this highly non-trivial?