Assuming $L$ is symmetric, elliptic second order differential operator, I want to to know about solutions to $$ -Lu = \lambda u \quad \text{in } \mathbf{R}^n.$$
Due to the unboundedness of the domain and the lack of boundary conditions, the classical theory on eigenvalue problems for elliptic PDE does not apply. My question is if there exists a theory of these kind of problems for distributions? The obvious example of this is on $\mathbf{R}$, with $-Lu = u''$, and $u(x) = \exp(i\sqrt{i\lambda x})$, since then for $\lambda > 0$, $u$ solves the eigenvalue problem $-Lu = \lambda u$, and $u$ is a tempered distribution. But I have not been able to find a general theory on this stuff, so any suggestions would be super!