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

We prove a Lieb--Thirring inequality for Schrödinger 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.