# The Lieb-Thirring inequalities: Recent results and open problems

@article{Frank2020TheLI, title={The Lieb-Thirring inequalities: Recent results and open problems}, author={Rupert L. Frank}, journal={arXiv: Mathematical Physics}, year={2020} }

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…

## 10 Citations

Lieb-Thirring inequalities and other functional inequalities for orthonormal systems

- Mathematics, Physics
- 2021

We review recent results on functional inequalities for systems of orthonormal functions. The key finding is that for various operators the orthonormality leads to a gain over a simple application of…

Two Consequences of Davies’ Hardy Inequality

- MathematicsFunctional Analysis and Its Applications
- 2021

Abstract Davies’ version of the Hardy inequality gives a lower bound for the Dirichlet integral of a function vanishing on the boundary of a domain in terms of the integral of the squared function…

Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori

- Mathematics, Physics
- 2021

In a previous paper we established Cwikel-type estimates and the CLR inequality for noncommutative tori. In this follow-up paper we extend these results to pseudodifferential operators and to curved…

A remark on the discrete spectrum of non-self-adjoint Jacobi operators

- Mathematics
- 2021

We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on…

Exchange and exclusion in the non-abelian anyon gas

- Physics, Mathematics
- 2020

We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on…

Some minimization problems for mean field models with competing forces

- Mathematics, Physics
- 2021

We review recent results on three families of minimization problems, defined on subsets of nonnegative functions with fixed integral. The competition between attractive and repulsive forces leads to…

Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori

- Mathematics
- 2021

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension n ≥ 2. We use them to derive a Cwikel-Lieb-Rozenblum inequality for the number of negative eigenvalues of…

Perturbation determinants and discrete spectra of semi-infinite non-self-adjoint Jacobi operators

- Mathematics
- 2021

We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on…

Direct methods to Lieb-Thirring kinetic inequalities

- Physics, Mathematics
- 2020

We review some recent progress on Lieb-Thirring inequalities, focusing on direct methods to kinetic estimates for orthonormal functions and applications for many-body quantum systems.

The periodic Lieb–Thirring inequality

- Physics, Mathematics
- 2020

We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the…

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