Improved sharp spectral inequalities for Schrödinger operators on the semi-axis

Abstract

We prove a Lieb-Thirring inequality for Schrödinger operators on the semi-axis with Robin boundary condition at the origin. The result improves on a bound obtained by P. Exner, A. Laptev and M. Usman [Commun. Math. Phys. 362(2), 531-541 ( 2014)]. The main difference in our proof is that we use the double commutation method in place of the single commutation method. We also establish an improved inequality in the case of a Dirichlet boundary condition.

Comparison bounds for perturbed Schrödinger operators with single-well potentials

“…Following its use by Benguria and Loss for Lieb-Thirring inequalities it has been employed to obtain analogous bounds for the Robin boundary case on the half-line by Exner, Laptev and Usman in [ELU14]. This was improved by Schimmer in [Sch19] by using an alteration of this idea, known as the double commutation method. Most recently, in [Lap21] the author used a variation of the factorisation method to derive the Hardy-Lieb-Thirring inequality, with a conjectured sharp constant.…”