On the ground state of lattice Schrödinger operators
Michal Jex
Czech Technical University in Prague, Praha, Czech RepublicFrantišek Štampach
Czech Technical University in Prague, Praha, Czech Republic

Abstract
We prove necessary and sufficient conditions for lattice Schrödinger operators to have a zero-energy bound state in arbitrary dimension. The two criteria are sharp, complementary, and depend crucially on both the dimension and asymptotic behaviour of the potential. The method relies on a discrete variant of Agmon’s comparison principle which is also proven. Our results represent a discrete variant of the recent criteria obtained in the continuous setting by D. Hundertmark, M. Jex, and M. Lange [Forum Math. Sigma 11 (2023), artile no. e61].
Cite this article
Michal Jex, František Štampach, On the ground state of lattice Schrödinger operators. J. Spectr. Theory 15 (2025), no. 2, pp. 647–678
DOI 10.4171/JST/558