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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