I've been studying Linear Time-Invariant (LTI) systems and came across the concept of eigenfunctions with respect to these systems. The literature often presents complex exponentials (as well as sines and cosines, by Euler's formula) as eigenfunctions of LTI systems. I understand that for an LTI system described by the operator $( \mathcal{H})$, an eigenfunction $( f )$ with eigenvalue $( \lambda )$ satisfies:
$$\mathcal{H}f = \lambda f$$
And for the case of complex exponentials, we have:
$$\mathcal{H}e^{st} = H(s)e^{st}$$
where $$H(s) $$ is the transfer function of the system.
However, is there a formal proof that the complex exponential is the only eigenfunction of such systems? Or are there other functions that can also satisfy the eigenfunction condition for LTI systems? If possible, I would appreciate a detailed explanation or a direction towards resources that tackle this proof.