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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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