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abstract: The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $$N$$ lowest eigenvalues of a Schr\odinger operator $$-\Delta-V(x)$ in terms of an $$L^p(\mathbb{R}^d)$$ norm of the potential $$V$$. We prove here the existence of an optimizing potential for each $$N$$, discuss its qualitative properties and the Euler--Lagrange equation (which is a system of coupled nonlinear Schr\odinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition $$\gamma>\max\{0,2-d/2\}$ on the Riesz exponent in the inequality, we prove the compactness of all the optimizing sequences up to translations. We also show that the optimal Lieb-Thirring constant cannot be stationary in $$N$$, which sheds a new light on a conjecture of Lieb-Thirring. In dimension $d=1$ at $$\gamma=3/2$$, we show that the optimizers with $$N$$ negative eigenvalues are exactly the Korteweg-de Vries $$N$$-solitons and that optimizing sequences must approach the corresponding manifold. Our work also covers the critical case $$\gamma=0$$ in dimension $$d\geq3$$ (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem. 

DOI: 10.4171/ECR/18-1/8 

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.