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- W3199516103 abstract "The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $N$ lowest eigenvalues of a Schrodinger 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 Schrodinger 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 covers the critical case $gamma=0$ in dimension $dgeq3$ (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem." @default.
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- W3199516103 date "2021-10-08" @default.
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- W3199516103 title "Optimizers for the finite-rank Lieb-Thirring inequality" @default.
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