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.
Author details: | Siegfried BeckusORCiDGND, Yehuda PinchoverORCiDGND |
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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 |