Quantum Systems at The Brink -- Existence and Decay Rates of Bound States at Thresholds; Critical Potentials and dimensionality

Dirk Hundertmark, Michal Jex, Markus Lange

Published 2021-07-29Version 1

One of the crucial properties of a quantum system is the existence of bound states. In this paper we present a necessary and sufficient condition for the Schr\"odinger operator to have a zero energy bound state. In particular we show that the asymptotic behaviour of the potential is the crucial ingredient. We derive the necessary and sufficient conditions for existence and absence with respect to the dimension. Our results are sharp and show high dependence on dimension.