Calogero Type Bounds in Two Dimensions

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