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

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

1995-04-17

arxiv.org/abs/quant-ph/9504013v1

Commun.Math.Phys. 178 (1996) 135-146

Physics

Quantum Physics

AMS-LATEX, 15 pages

Scientific paper

10.1007/BF02104912

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R} V^{\gamma+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<\gamma<3/2.

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