Calogero type bounds in two dimensions

For a Schrödinger operator on the plane R² with electric potential V and an Aharonov–Bohm magnetic field, we obtain an upper bound on the number of its negative eigenvalues in terms of the L¹(R²) -norm of V. Similar to Calogero’s bound in one dimension, the result is true under monotonicity assumptions on V. Our method of proof 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 Dirichlet boundary condition is imposed and on the whole plane when restricted to antisymmetric functions.