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

Simon Larson,Phan Thành Nam,Douglas Lundholm

doi:10.5802/ahl.72

Abstract

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 functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.

Highlights

In that context it is well known that stability of the first kind, i.e. that the ground state energy of the Coulomb system is finite, follows from some sort of the uncertainty principle (e.g. Sobolev’s inequality)
Lieb–Thirring inequalities for wave functions vanishing on the diagonal set more subtle: for this the fermionic nature of particles is crucial
Our proof is based on a general strategy of deriving Lieb–Thirring inequalities for wave functions satisfying some partial exclusion properties, which was proposed by Lundholm and Solovej in [LS13a] and developed further in [FS12, LL18, LNP16, LPS15, LS13b, LS14, LS18, Lun[18], Nam18]

Summary

Introduction

The celebrated Lieb–Thirring inequality states that the expected kinetic energy of a free Fermi gas is bounded from below by its semiclassical approximation up to a universal factor, namely (1.1). In that context it is well known that stability of the first kind, i.e. that the ground state energy of the Coulomb system is finite, follows from some sort of the uncertainty principle (e.g. Sobolev’s inequality). Lieb–Thirring inequalities for wave functions vanishing on the diagonal set more subtle: for this the fermionic nature of particles is crucial. The emergence of the factor N −2s/d can be seen by considering the bosonic trial state ΨN = u⊗N (whose density is ΨN (x) = N |u(x)|2). Note that Pauli’s exclusion principle (1.2) implies that the wave function ΨN vanishes on the diagonal set (1.4). — Does the Lieb–Thirring inequality (1.1) remain valid if the anti-symmetry assumption (1.2) is replaced by the weaker condition ΨN | = 0 ?. The precise statement of our result and its consequences will be presented

Main results

General strategy of deriving Lieb–Thirring inequalities

Reduction of local exclusion

Many-body Poincaré inequality