Calogero type bounds in two dimensions

Abstract

For a Schrödinger operator on the plane R 2 with electric potential V and Aharonov-Bohm magnetic field we obtain an upper bound on the number of its negative eigenvalues in terms of the L 1 pR 2 q-norm of V . Similar to Calogero's bound in one dimension, the result is true under monotonicity assumptions on V . Our proof method relies on a generalisation of Calogero's bound to operator-valued potentials. We also establish a similar bound for the Schrödinger operator (without magnetic field) on the half-plane when a Dirchlet boundary condition is imposed and on the whole plane when restricted to antisymmetric functions.