Improvements on the Lieb-Thirring and Cwikel-Lieb-Rozenblum bounds via a three lines lemma

Abstract: In this talk I will present some recent improvements [1,2] on the best-known upper bounds for the optimal constants in the Lieb-Thirring (LT) and Cwikel-Lieb-Rozenblum (CLR) inequalities. These improvements were obtained by explicitly solving an interesting family of variational problems, whose connection with the LT and CLR inequalities has recently been established in [3] and [4], respectively. To obtain the explicit solutions, we first reformulate these variational problems as an interpolation-like inequality along three parallel lines in the complex plane, for certain spaces of holomorphic functions. We then explicitly compute the optimizers for these interpolation inequalities by combining some techniques of complex, Fourier, and convex analysis.

References: [1] Corso and Ried - On a variational problem related to the Cwikel-LiebRozenblum and Lieb-Thirring inequalities - arxiv.org/abs/2403.04347 [2] Corso - A generalized three lines lemma in Hardy-like spaces - arxiv.org/abs/2407.10117 [3] Frank, Hundertmark, Jex, and Nam - The Lieb-Thirring inequality revisited - ems.press/journals/jems/articles/666342 [4] Hundertmark, Kunstmann, Ried, and Vugalter - Cwikel’s bound reloaded - link.springer.com/article/10.1007/s00222-022-01144-7

mathematical physicsspectral theory

Audience: researchers in the topic

Munich-Copenhagen-Santiago Mathematical Physics seminar

Series comments: The MAS-MP seminar series has now changed name to MCS-MP.

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