Shnol-type theorem for the Agmon ground state

Abstract

Let $H$ be a Schrödinger operator defined on a noncompact Riemannian manifold $Ω$, and let $W\in L^\infty(Ω;\mathbb{R})$. Suppose that the operator $H+W$ is critical in $Ω$, and let $φ$ be the corresponding Agmon ground state. We prove that if $u$ is a generalized eigenfunction of $H$ satisfying $|u|\leq φ$ in $Ω$, then the corresponding eigenvalue is in the spectrum of $H$. The conclusion also holds true if for some $K\Subset Ω$ the operator $H$ admits a positive solution in $\tildeΩ=Ω\setminus K$, and $|u|\leq ψ$ in $\tildeΩ$, where $ψ$ is a positive solution of minimal growth in a neighborhood of infinity in $Ω$. Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.

12 pages