The Lieb–Thirring inequalities: Recent results and open problems

Abstract

This review celebrates the generous gift by Ronald and Maxine Linde for the remodeling of the Caltech mathematics department and the author is very grateful to the editors of this volume for the invitation to contribute. We attempt to survey recent results and open problems connected to Lieb-Thirring inequalities. In view of several excellent existing reviews [132,17,113,91,108] as well as highly recommended textbooks [134,136], we sometimes put our focus on developments during the past decade.The author would like to thank all his collaborators on the topic of Lieb-Thirring inequalities and, in particular, A.

Schrödinger Operators with Complex Sparse Potentials

“…On the proof of Theorem 2. Even if V is not necessarily compactly supported anymore, we expect the decay of V to smoothen out the resolvents in Gelfand's formula (14). To quantify the decay of V , we apply a horizontal dyadic decomposition (see, e.g., [26,Theorem 6.6]), which is reminiscent to the definition of Lorentz spaces.…”

“…where, for fixed i, j, the V ijk are supported on a sparse collection of balls {B(x k , R i )} Ni k=1 . Plugging the decomposition (35) into Gelfand's formula (14) allows us to proceed similarly as in the case of compactly supported potentials. We refer to [9, Sections 7.2-7.4] for details.…”

“…complex-valued V , the analysis of these quantities yields results that are surprisingly different from those in the case of real-valued V . We refer to [14,Section 5.13] and the references therein for a survey.…”

The state of the Lieb–Thirring conjecture

“…We remark that there is by now a large literature on Hardy-Lieb-Thirring inequalities for (fractional) Laplacians where a positive Hardy weight is subtracted from the operator, initiated by Ekholm and Frank [12]. For a review, we refer to [17].…”

“…This review aims to focus on the current state of the subject matter. For a comprehensive review of Lieb-Thirring inequalities including recent research directions we refer to Frank's [17]. We also mention Hundertmark's review [31] which similarly provides a summary of the state of the field in 2007.…”