The Lieb–Thirring inequality revisited

@article{Frank2018TheLI,
  title={The Lieb–Thirring inequality revisited},
  author={Rupert L. Frank and Dirk Hundertmark and Michal Jex and Phan Th{\`a}nh Nam},
  journal={Journal of the European Mathematical Society},
  year={2018}
}
We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the negative eigenvalues of Schr\"odinger operators, which significantly improve the so far existing bounds. 
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References

SHOWING 1-10 OF 23 REFERENCES
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.
Lieb-Thirring inequalities with improved constants
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
Sharp Lieb-Thirring inequalities in high dimensions
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 $L^{cl}_{\gamma,d}$
A positive density analogue of the Lieb-Thirring inequality
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
Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator
There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given Lp norm of the potential. We give effective estimates for the amount by
Cwikel's theorem and the CLR inequality
We give a short proof of the Cwikel-Lieb-Rozenblum (CLR) bound on the number of negative eigenvalues of Schr\"odinger operators. The argument, which is based on work of Rumin, leads to remarkably
On the number of bound states for Schrödinger operators with operator-valued potentials
Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schrödinger operators with
Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces
Abstract We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting in L 2 ( Ω ) (or L 2 ( Ω ,  C m
...
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