On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2

Abstract

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schrödinger operator $Hu:=-u^{\prime \prime }-Vu,V\ge 0,$ on $L_2(\mathbb{R})$. We prove the inequality \sum_i|E_i(H)|^γ\leq L_{γ,1}\int_{\Bbb R} V^{γ+1/2}(x)dx, (1) for the "limit" case $\gamma =1/2.$ This will imply improved estimates for the best constants $L_{\gamma ,1}$ in (1), as $1/2<γ<3/2.

AMS-LATEX, 15 pages

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