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Shnol-type theorem for the Agmon ground state

  • LetH be a Schrodinger operator defined on a noncompact Riemannianmanifold Omega, and let W is an element of L-infinity (Omega; R). Suppose that the operator H + W is critical in Omega, and let phi be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction ofH satisfying vertical bar u vertical bar <= C-phi in Omega for some constant C > 0, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K is an element of Omega the operator H admits a positive solution in (Omega) over bar = Omega \ K, and vertical bar u vertical bar <= C psi in (Omega) over bar for some constant C > 0, where psi is a positive solution of minimal growth in a neighborhood of infinity in Omega. Under natural assumptions, this result holds also in the context of infinite graphs, and Dirichlet forms.

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Metadaten
Author details:Siegfried BeckusORCiDGND, Yehuda PinchoverORCiDGND
DOI:https://doi.org/10.4171/JST/296
ISSN:1664-039X
ISSN:1664-0403
Title of parent work (English):Journal of spectral theory
Publisher:EMS Publishing House
Place of publishing:Zürich
Publication type:Article
Language:English
Date of first publication:2020/02/28
Publication year:2020
Release date:2023/03/21
Tag:Caccioppoli inequality; Schrodinger operators; Shnol theorem; generalized eigenfunction; graphs; ground state; positive solutions; weighted
Volume:10
Issue:2
Number of pages:23
First page:355
Last Page:377
Funding institution:Israel Science FoundationIsrael Science Foundation [970/15]
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
DDC classification:5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik
Peer review:Referiert
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