# 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)

## 170 Citations

### Lieb-Thirring Inequalities for Geometrically Induced Bound States

- Mathematics
- 2004

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schrödinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To…

### Lieb-Thirring inequalities with improved constants

- Mathematics
- 2007

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…

### Connection Between the Lieb-Thirring Conjecture for Schrödinger Operators and An Isoperimetric Problem for Ovals on the Plane

- Mathematics
- 2004

To determine the sharp constants for the one dimensional Lieb– Thirring inequalities with exponent γ ∈ (1/2, 3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp…

### Connection between the Lieb--Thirring conjecture for Schroedinger operators and an isoperimetric problem for ovals on the plane

- Mathematics
- 2004

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…

### New constants in discrete Lieb–Thirring inequalities for Jacobi matrices

- Mathematics
- 2010

As is known, some results used for improving constants in the Lieb–Thirring inequalities for Schrödinger operators in L2(−∞, ∞) can be translated to discrete Schrödinger operators and, more…

### Lieb-Thirring inequalities for higher order differential operators

- Mathematics
- 2004

A BSTRACT . We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order γ ≥ 3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to…

### Universal monotonicity of eigenvalue moments and sharp Lieb-Thirring inequalities

- Mathematics
- 2008

We show that phase space bounds on the eigenvalues of Schrodinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with…

### Lieb-Thirring inequalities on the torus

- Mathematics
- 2015

We consider the Lieb-Thirring inequalities on the -dimensional torus with arbitrary periods. In the space of functions with zero average with respect to the shortest coordinate we prove the…

### Lieb–Thirring inequalities for higher order differential operators

- Mathematics
- 2004

We derive Lieb–Thirring inequalities for the Riesz means of eigenvalues of order γ ≥ 3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic…

### A positive density analogue of the Lieb-Thirring inequality

- Mathematics
- 2011

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schr\"odinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system…

## References

SHOWING 1-10 OF 37 REFERENCES

### On Inequalities for the Bound States of Schrödinger Operators

- Mathematics
- 1995

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

- Mathematics
- 1995

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

- Mathematics
- 2000

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

- Mathematics
- 1998

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

- Mathematics
- 1990

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

- Mathematics
- 1977

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

- Mathematics
- 1984

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

- Mathematics
- 1976

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

- Mathematics
- 1996

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…