## On the Lieb-Thirring constants $$L_{\gamma, 1}$$ for $$\gamma \geq 1/2$$.(English)Zbl 0858.34075

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

### MSC:

 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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### References:

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