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On the Lieb-Thirring constantsL γ,1 for γ≧1/2

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Abstract

LetE i(H) denote the negative eigenvalues of the one-dimensional Schrödinger operatorHu≔−u″−Vu,V≧0, onL 2(∝). We prove the inequality

$$\mathop \sum \limits_i |E_i (H)|^{ \gamma } \leqq L_{\gamma ,1} \mathop \smallint \limits_\mathbb{R} V^{\gamma + 1/2} (x)dx,$$
((1))

for the “limit” case γ=1/2. This will imply improved estimates for the best constantsL γ,1 in (1) as 1/2<γ<3/2.

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Weidl, T. On the Lieb-Thirring constantsL γ,1 for γ≧1/2. Commun.Math. Phys. 178, 135–146 (1996). https://doi.org/10.1007/BF02104912

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