Sharp Lieb-Thirring inequalities in high dimensions

@article{Laptev1999SharpLI,
  title={Sharp Lieb-Thirring inequalities in high dimensions},
  author={Ari Laptev and Timo Weidl},
  journal={Acta Mathematica},
  year={1999},
  volume={184},
  pages={87-111}
}
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 $L^{cl}_{\gamma,d}$ with $\gamma\ge 3/2$ and arbitrary $d\ge 1$. (revised, to appear in Acta Math) 

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References

SHOWING 1-10 OF 37 REFERENCES

On Inequalities for the Bound States of Schrödinger Operators

We improve the Lieb constant in the Cwikel-Lieb-Rozenblum inequality for the number of bound states of Schrodinger operators whose potential equals the characteristic function of a measurable set.

On the Lieb-Thirring constantsLγ,1 for γ≧1/2

AbstractLetEi(H) denote the negative eigenvalues of the one-dimensional Schrödinger operatorHu≔−u″−Vu,V≧0, onL2(∝). We prove the inequality(1) $$\mathop \sum \limits_i |E_i (H)|^{ \gamma } \leqq

New bounds on the Lieb-Thirring constants

Abstract.Improved estimates on the constants Lγ,d, for 1/2<γ<3/2, d∈N, in the inequalities for the eigenvalue moments of Schrödinger operators are established.

A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator

We give a proof of the Lieb-Thirring inequality in the critical case d=1, γ = 1/2, which yields the best possible constant.

Riesz means of bounded states and semi-classical limit connected with a Lieb-Thirring conjecture II

On etudie le probleme des valeurs propres negatives de l'operateur de Schrodinger avec un c ∞ -potentiel d'interaction. On compare la constante universelle obtenue par Lieb et Thirring a celle de la

Bounds on the number of eigenvalues of the Schrödinger operator

Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:i)A connection between the moments of order (n − 1)/2

On characteristic exponents in turbulence

Ruelle has found upper bounds to the magnitude and to the number of non-negative characteristic exponents for the Navier-Stokes flow of an incompressible fluid in a domain Θ. The latter is

Bounds on the eigenvalues of the Laplace and Schroedinger operators

If 12 is an open set in R", and if N(£l, X) is the number of eigenvalues of A (with Dirichlet boundary conditions on d£2) which are < X (k > 0), one has the asymptotic formula of Weyl [1] , [2] : l i

On the Lieb-Thirring constants L ?,1 for ??1/2

Let E i (H) denote the negative eigenvalues of the one-for the "limit" case = 1=2: This will imply improved estimates for the best constants L ;1 in (1) as 1=2 < < 3=2: 0. Let H = ??V denote the