Let H be a generalized Schrödinger operator on a domain of a non-compact connected Riemannian manifold, and a generalized eigenfunction u for H: that is, u satisfies the equation Hu = λu in the weak sense but is not necessarily in L2. The problem is to find conditions on the growth of u, so that λ belongs to the spectrum of H. We unify and generalize known results on this problem. In addition, a variety of examples is provided, illustrating the different nature of the growth conditions.
