TY  - JOUR
A1  - Beckus, Siegfried
A1  - Pinchover, Yehuda
T1  - Shnol-type theorem for the Agmon ground state
JF  - Journal of spectral theory
N2  - 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.
KW  - Shnol theorem
KW  - Caccioppoli inequality
KW  - Schrodinger operators
KW  - generalized eigenfunction
KW  - ground state
KW  - positive solutions
KW  - weighted
KW  - graphs
Y1  - 2020
U6  - https://doi.org/10.4171/JST/296
SN  - 1664-039X
SN  - 1664-0403
VL  - 10
IS  - 2
SP  - 355
EP  - 377
PB  - EMS Publishing House
CY  - Zürich
ER  - 
TY  - JOUR
A1  - Beckus, Siegfried
A1  - Eliaz, Latif
T1  - Eigenfunctions growth of R-limits on graphs
JF  - Journal of spectral theory / European Mathematical Society
N2  - A characterization of the essential spectrum of Schrodinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z(d) as "right-limits," captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in sigma(ss)(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.
KW  - Essential spectrum
KW  - Schrodinger operators
KW  - graphs
KW  - right limits
KW  - generalized eigenfunctions
Y1  - 2021
U6  - https://doi.org/10.4171/JST/389
SN  - 1664-039X
SN  - 1664-0403
VL  - 11
IS  - 4
SP  - 1895
EP  - 1933
PB  - EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut für Mathematik, Technische Universität
CY  - Berlin
ER  -