Improved sharp spectral inequalities for Schr\"odinger operators on the semi-axis

TL;DR

This paper establishes improved spectral inequalities for Schrödinger operators on the semi-axis with Robin and Dirichlet boundary conditions, using advanced commutation methods to refine previous bounds under certain integrability assumptions.

Contribution

It introduces a novel proof technique using double commutation to improve Lieb--Thirring inequalities for semi-axis Schrödinger operators with boundary conditions.

Findings

Improved Lieb--Thirring bounds for Robin boundary conditions.

Enhanced inequalities for Dirichlet boundary conditions.

Use of double commutation method for sharper spectral estimates.

Abstract

We prove a Lieb--Thirring inequality for Schr\"odinger operators $-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+V$ 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)] albeit under the additional assumption $V\in L^1(\mathbb{R}_+)$. 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.