On the ground state of lattice Schrödinger operators

  • Michal Jex

    Czech Technical University in Prague, Praha, Czech Republic
  • František Štampach

    Czech Technical University in Prague, Praha, Czech Republic
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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