Calogero type bounds in two dimensions

TL;DR

This paper derives bounds on the number of negative eigenvalues for 2D Schrödinger operators with electric and magnetic potentials, extending Calogero's bounds to new settings with monotonicity assumptions.

Contribution

It generalizes Calogero's eigenvalue bounds to two-dimensional Schrödinger operators with magnetic fields and operator-valued potentials, under monotonicity conditions.

Findings

Bound on negative eigenvalues in terms of $L^1$-norm of potential

Extension of bounds to magnetic Schrödinger operators on the plane

Bound for Schrödinger operators on the half-plane with Dirichlet boundary

Abstract

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 on the number of its negative eigenvalues in terms of the $L^1(\mathbb{R}^2)$-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 Schr\"odinger 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.