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Let $H$ be a generalized Schr\"odinger operator on a domain of a non-compactconnected Riemannian manifold, and a generalized eigenfunction $u$ for $H$:that is, $u$ satisfies the equation $Hu=\lambda u$ in the weak sense but is notnecessarily in $L^2$. The problem is to find conditions on the growth of $u$,so that $\lambda$ belongs to the spectrum of $H$. We unify and generalize knownresults on this problem. In addition, a variety of examples is provided,illustrating the different nature of the growth conditions.

Generalized eigenfunctions and eigenvalues: a unifying framework for Shnol-type theorems