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Sharp Lieb-Thirring inequalities in high dimensions. (English) Zbl 1142.35531

From the introduction: Let us consider a Schrödinger operator in \(L^2(\mathbb R^d)\), \(-\Delta+V\), where \(V\) is a real-valued function. E. H. Lieb and W. E. Thirring [Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Stud. math. Phys., Essays Honor Valentine Bargmann, 269–303 (1976; Zbl 0342.35044)] proved that if \(\gamma>\max(0,1-\frac12 d)\), then there exist universal constants \(L_{\gamma,d}\) satisfying
\[ \text{tr}(-\Delta+V)_-^\gamma\leq L_{\gamma,d} \int_{\mathbb R^d} V_-^{\gamma+d/2}(x)\,dx. \]
The main purpose of this paper is to verify \(L_{\gamma,d}= L_{\gamma,d}^{\text{cl}}\) for any \(\gamma\geq \frac32\), \(d\in\mathbb N\) and any \(V\in L^{\gamma+d/2}(\mathbb R^d)\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
47F05 General theory of partial differential operators
81U05 \(2\)-body potential quantum scattering theory

Citations:

Zbl 0342.35044
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References:

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