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

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schrödinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. 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 $γ=1/2.$ This will imply improved estimates for the best constants $L_{γ,1}$ in (1), as $1/2<γ<3/2.

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