Thiago Carvalho Corso (University of Stuttgart)

Title: The Cwikel-Lieb-Rozenblum inequality and a variant of the Hadamard three lines lemma
Abstract: The celebrated Cwikel-Lieb-Rozenblum (CLR) inequality, which gives a semiclassical bound on the number of bound states of a Schr\"odinger operator, is by now one of the cornerstones of quantum mechanics. Finding the sharp constant in the CLR inequality, however, remains an important open problem in mathematical physics. Recently, some progress has been made by Hundertmark, Kunstmann, Ried, and Vugalter in Invent. math. (2023) 231:111–167 [1]. There, the authors derived a variational problem for obtaining upper bounds on the CLR constant, from which they obtained the first improvement on the constants derived by Lieb more than 40 years ago, for dimensions d>=5. Still, the question of whether their method yields the sharp constant or is able to improve on Lieb's upper bound for d=3 (physically the most relevant case) and d=4 remained open. In this talk, we present a reformulation of the variational problem derived in [1] that resembles the variational problem associated to the famous Hadamard three lines lemma. This reformulation admits an unique (up to symmetries) optimizer for which we provide an explicit analytic formula that can be numerically evaluated. As a consequence, we improve the upper bounds on the CLR constant derived in [1] for dimensions d>=5, and show that their approach is not able to improve over Lieb's constant for dimensions d=3 and d=4.