# Shnol-type theorem for the Agmon ground state

#### Autoren: Siegfried Beckus, Yehuda Pinchover (2018)

Let H be a Schrödinger operator defined on a noncompact Riemannian manifold Ω, and let W∈L∞(Ω;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|≤φ in Ω, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K⋐Ω the operator H admits a positive solution in Ω'=Ω∖K, and |u|≤ψ in Ω', 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.

Zeitschrift:
Journal of Spectral Theory
Reihe:
10
Seiten:
355 - 377

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