• Corpus ID: 220646446

The Lieb-Thirring inequalities: Recent results and open problems

  title={The Lieb-Thirring inequalities: Recent results and open problems},
  author={Rupert L. Frank},
  journal={arXiv: Mathematical Physics},
  • R. Frank
  • Published 18 July 2020
  • Physics, Mathematics
  • arXiv: Mathematical Physics
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… 
Lieb-Thirring inequalities and other functional inequalities for orthonormal systems
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
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
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
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
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
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
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
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
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
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


Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems
This paper is devoted to inequalities of Lieb-Thirring type. Let V be a nonnegative potential such that the corresponding Schrodinger operator has an unbounded sequence of eigenvalues (λi(V ))i∈N∗.
Sharp Lieb-Thirring inequalities in high dimensions
We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{\gamma,d}$
The orthonormal Strichartz inequality on torus
  • Shohei Nakamura
  • Mathematics
    Transactions of the American Mathematical Society
  • 2019
In this paper, motivated by recent works due to Frank-Lewin-Lieb-Seiringer and Frank-Sabin, we study the Strichartz inequality on torus with the orthonormal system input and obtain sharp estimates in
A Simple Proof of Hardy-Lieb-Thirring Inequalities
We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schrödinger operators. The proof covers the optimal parameter range. It is based on a
Best constants in Lieb-Thirring inequalities: a numerical investigation
We investigate numerically the optimal constants in Lieb-Thirring inequalities by studying the associated maximization problem. We use a monotonic fixed-point algorithm and a finite element
Connection between the Lieb--Thirring conjecture for Schroedinger operators and an isoperimetric problem for ovals on the plane
To determine the sharp constants for the one dimensional Lieb--Thirring inequalities with exponent gamma in (1/2,3/2) is still an open problem. According to a conjecture by Lieb and Thirring the
Geometrical Versions of improved Berezin-Li-Yau Inequalities
We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$,
Lieb-Thirring inequalities with improved constants
Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring
Cwikel's theorem and the CLR inequality
We give a short proof of the Cwikel-Lieb-Rozenblum (CLR) bound on the number of negative eigenvalues of Schr\"odinger operators. The argument, which is based on work of Rumin, leads to remarkably
Lieb-Thirring inequalities for higher order differential operators
We derive Lieb-Thirring inequalities for the Riesz means of eigen-values of order γ≥3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic