# Shnol-type theorem for the Agmon ground state

## TL;DR

This paper extends Shnol-type theorems to Schrödinger operators on Riemannian manifolds, graphs, and Dirichlet forms, showing that eigenfunctions dominated by ground states imply spectral inclusion.

## Contribution

It proves a generalized Shnol-type theorem for critical Schrödinger operators on manifolds, graphs, and Dirichlet forms, linking eigenfunction bounds to spectral properties.

## Key findings

- Eigenfunctions bounded by Agmon ground states imply spectral inclusion.
- The result applies to operators on manifolds, graphs, and Dirichlet forms.
- The theorem extends classical results to more general geometric and discrete settings.

## Abstract

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