## 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

Zbl 0342.35044
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### References:

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