A Simple Proof of Hardy-Lieb-Thirring Inequalities

“…The statement follows from Lemma 20 by the usual arguments (see e.g. the proof of Theorem 1.1 in [4]): Let γ > 0 and q ∈ (1, 1 + γ). By the minimax principle we have…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…In the cases marked with "-" there exists no self-adjoint realisation, see Theorem 1. Note that in the case "Ia1" inequality (1.6) is a form of the HardyLieb-Thirring inequality (see [3,4,7]). …”

Section: Introductionmentioning

confidence: 99%

On the Virtual Levels of Positively Projected Massless Coulomb–Dirac Operators

Morozov

Muller

2017

Ann. Henri Poincaré

View full text Add to dashboard Cite

Considering different self-adjoint realisations of positively projected massless Coulomb-Dirac operators we find out, under which conditions any negative perturbation, however small, leads to emergence of negative spectrum. We also prove some weighted Lieb-Thirring estimates for negative eigenvalues of such operators. In the process we find explicit spectral representations for all self-adjoint realisations of massless Coulomb-Dirac operators on the half-line.

show abstract

“…This particular feature turns out to be extremely convenient, and sidesteps important complications that would arise had we used an estimate with a non-local error: the reader here should compare (1.27) with estimates where the error is nonlocal: see, for example, an exact formula due to M. Loss (1.27) allows us in particular to split the trace of the negative part of |σ · p A | 2s − C s /|x| 2s + V as the sum of that of two localized Hamiltonians -one which is localized around the origin and one far away from it. For the first Hamiltonian we apply Estimate (1.26) and then apply the Hardy-Lieb-Thirring inequality (1.9) as appears, for example, in [14]. For the part far away we obtain an effective localized potential, roughly equal to −Eφ/|x| 2s + V , with E a constant that goes beyond C s and φ the indicator function of a region far from the origin.…”

Section: The Strategy Of Proof and The Main Tools Usedmentioning

confidence: 99%

Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

Bley

Fournais

2018

Commun. Math. Phys.

View full text Add to dashboard Cite

We provide lower bounds for the sum of the negative eigenvalues of the operator |σ · pA| 2s − Cs/|x| 2s + V in three dimensions, where s ∈ (0, 1], covering the interesting physical cases s = 1 and s = 1/2. Here σ is the vector of Pauli matrices, pA = p − A, with p = −i∇ the three-dimensional momentum operator and A a given magnetic vector potential, and Cs is the critical Hardy constant, that is, the optimal constant in the Hardy inequality |p| 2s ≥ Cs/|x| 2s . If spin is neglected, results of this type are known in the literature as Hardy-Lieb-Thirring inequalities, which bound the sum of negative eigenvalues from below by −Ms V 1+3/(2s) − , for a positive constant Ms. The inclusion of magnetic fields in this case follows from the non-magnetic case by diamagnetism. The addition of spin, however, offers extra challenges that make the result more elusive. It is the purpose of this article to resolve this problem by providing simple bounds for the sum of the negative eigenvalues of the operator in question. In particular, for 1/2 ≤ s ≤ 1 we are able to express the bound purely in terms of the magnetic field energy B 2 2 and integrals of powers of the negative part of V .

show abstract

A Simple Proof of Hardy-Lieb-Thirring Inequalities

Cited by 55 publications

References 24 publications

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

Lieb–Thirring and Cwickel–Lieb–Rozenblum inequalities for perturbed graphene with a Coulomb impurity

On the Virtual Levels of Positively Projected Massless Coulomb–Dirac Operators

Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

Contact Info

Product

Resources

About