Counterexample to the Laptev--Safronov conjecture

TL;DR

This paper disproves the Laptev--Safronov conjecture in certain cases, providing a counterexample and establishing new sharp bounds for a broad class of Schrödinger operators.

Contribution

It presents a counterexample to the conjecture and extends the results to a large class of Schrödinger-type operators with new sharp bounds.

Findings

Counterexample disproves the conjecture in some ranges

New sharp upper bounds for Schrödinger-type operators

Counterexample adaptable to various operators

Abstract

We prove that the Laptev--Safronov conjecture (Comm. Math. Phys., 2009) is false in the range that is not covered by Frank's positive result (Bull. Lond. Math. Soc., 2011). The simple counterexample is adaptable to a large class of Schr\"odinger type operators, for which we also prove new sharp upper bounds.