# Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems

@article{Lundholm2015FractionalHA, title={Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems}, author={Douglas Lundholm and Phan Th{\`a}nh Nam and Fabian Portmann}, journal={Archive for Rational Mechanics and Analysis}, year={2015}, volume={219}, pages={1343-1382} }

We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.

## 28 Citations

### Lieb–Thirring inequalities for wave functions vanishing on the diagonal set

- MathematicsAnnales Henri Lebesgue
- 2021

We propose a general strategy to derive Lieb-Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb-Thirring inequality to wave…

### EULER–LAGRANGE EQUATIONS ASSOCIATED WITH EXTREMAL FUNCTIONS OF SEVERAL NONLOCAL INEQUALITIES

- Mathematics
- 2020

This paper is concerned with the extremal functions of several kinds of non-local inequalities, including the Hardy-Littlewood-Sobolev inequality, fractional Gagliardo-Nirenberg inequality, nonlocal…

### Direct methods to Lieb-Thirring kinetic inequalities

- 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.

### One-dimensional Lieb-Oxford bounds.

- Mathematics, PhysicsThe Journal of chemical physics
- 2020

This work investigates and proves Lieb-Oxford bounds in one dimension by studying convex potentials that approximate the ill-defined Coulomb potential by utilizing logarithmic expressions of the particle density.

### A Lieb-Thirring inequality for extended anyons

- Physics, Mathematics
- 2022

. We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. That is, we consider 2D fermionic particles coupled to…

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

- Economics
- 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…

### Lieb-Thirring Bounds for Interacting Bose Gases

- Physics
- 2015

We study interacting Bose gases and prove lower bounds for the kinetic plus interaction energy of a many-body wave function in terms of its particle density. These general estimates are then applied…

### Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Sobolev-Coulomb spaces

- Mathematics
- 2021

Abstract. We establish the full range Gagliardo-Nirenberg and the Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Sobolev-Coulomb spaces for the (fractional) derivative 0 ≤ s ≤…

### Regularity of many-body Schr\"odinger evolution equation and its application to numerical analysis

- Mathematics
- 2018

A decade ago, the mixed regularity of stationary many-body Schr\"o\-dinger equation has been studied by Harry Yserentant through the Pauli Principle and the Hardy inequality (Uncertainty Principle).…

## References

SHOWING 1-10 OF 55 REFERENCES

### A Simple Proof of Hardy-Lieb-Thirring Inequalities

- Mathematics
- 2009

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…

### Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

- Mathematics
- 2006

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrodinger-like operators remain true, with possibly different constants, when the critical Hardy-weight C │x│^(-2)…

### On Lieb-Thirring Inequalities for Schrödinger Operators with Virtual Level

- Mathematics
- 2006

We consider the operator H=−Δ−V in L2(ℝd), d≥3. For the moments of its negative eigenvalues we prove the estimate Similar estimates hold for the one-dimensional operator with a Dirichlet condition at…

### Local Exclusion and Lieb–Thirring Inequalities for Intermediate and Fractional Statistics

- Mathematics
- 2013

In one and two spatial dimensions there is a logical possibility for identical quantum particles different from bosons and fermions, obeying intermediate or fractional (anyon) statistics. We consider…

### Hardy and Lieb-Thirring Inequalities for Anyons

- Mathematics
- 2013

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter $${\alpha \in [0, 1]}$$α∈[0,1] ranging from bosons…

### Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems

- Mathematics
- 2013

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy.…

### Lieb-Thirring Inequality for a Model of Particles with Point Interactions

- Physics
- 2012

We consider a model of quantum-mechanical particles interacting via point interactions of infinite scattering length. In the case of fermions we prove a Lieb-Thirring inequality for the energy, i.e.,…

### Lieb-Thirring Bounds for Interacting Bose Gases

- Physics
- 2015

We study interacting Bose gases and prove lower bounds for the kinetic plus interaction energy of a many-body wave function in terms of its particle density. These general estimates are then applied…

### Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator

- Mathematics
- 2011

We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for…