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  - Keller, Matthias
A1  - Pinchover, Yehuda
A1  - Pogorzelski, Felix
T1  - From hardy to rellich inequalities on graphs
JF  - Proceedings of the London Mathematical Society
N2  - We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
KW  - 35R02
KW  - 39A12 (primary)
KW  - 26D15
KW  - 31C20
KW  - 35B09
KW  - 58E35 (secondary)
Y1  - 2020
U6  - https://doi.org/10.1112/plms.12376
SN  - 0024-6115
SN  - 1460-244X
VL  - 122
IS  - 3
SP  - 458
EP  - 477
PB  - Wiley
CY  - Hoboken
ER  - 
TY  - GEN
A1  - Keller, Matthias
A1  - Pinchover, Yehuda
A1  - Pogorzelski, Felix
T1  - From hardy to rellich inequalities on graphs
T2  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1379 
KW  - 35R02
KW  - 39A12 (primary)
KW  - 26D15
KW  - 31C20
KW  - 35B09
KW  - 58E35 (secondary)
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-542140
SN  - 1866-8372
IS  - 3
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Pinchover, Yehuda
A1  - Pogorzelski, Felix
T1  - Criticality theory for Schrödinger operators on graphs
JF  - Journal of spectral theory
N2  - We study Schrodinger operators given by positive quadratic forms on infinite graphs. From there, we develop a criticality theory for Schrodinger operators on general weighted graphs.
KW  - green function
KW  - ground state
KW  - positive solutions
KW  - discrete Schrodinger
KW  - operators
KW  - weighted graphs
Y1  - 2019
U6  - https://doi.org/10.4171/JST/286
SN  - 1664-039X
SN  - 1664-0403
VL  - 10
IS  - 1
SP  - 73
EP  - 114
PB  - European Mathematical Society
CY  - Zürich
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Pinchover, Yehuda
A1  - Pogorzelski, Felix
T1  - Optimal Hardy inequalities for Schrodinger operators on graphs
JF  - Communications in mathematical physics
N2  - For a given subcritical discrete Schrodinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H - lambda w is subcritical in X for all lambda < 1, null-critical in X for lambda = 1, and supercritical near any neighborhood of infinity in X for any lambda > 1. Our results rely on a criticality theory for Schrodinger operators on general weighted graphs.
Y1  - 2018
U6  - https://doi.org/10.1007/s00220-018-3107-y
SN  - 0010-3616
SN  - 1432-0916
VL  - 358
IS  - 2
SP  - 767
EP  - 790
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Pinchover, Yehuda
A1  - Pogorzelski, Felix
T1  - An improved discrete hardy inequality
JF  - The American mathematical monthly : an official publication of the Mathematical Association of America
N2  - In this note, we prove an improvement of the classical discrete Hardy inequality. Our improved Hardy-type inequality holds with a weight w which is strictly greater than the classical Hardy weight w(H)(n) : 1/(2n)(2), where N.
KW  - Primary 26D15
Y1  - 2018
U6  - https://doi.org/10.1080/00029890.2018.1420995
SN  - 0002-9890
SN  - 1930-0972
VL  - 125
IS  - 4
SP  - 347
EP  - 350
PB  - Taylor & Francis Group
CY  - Philadelphia
ER  -