Related papers: Direct methods to Lieb-Thirring kinetic inequaliti…
We review recent results on functional inequalities for systems of orthonormal functions. The key finding is that for various operators the orthonormality leads to a gain over a simple application of the triangle inequality. The operators…
We attempt to survey recent results and open problems connected to Lieb-Thirring inequalities.
This is a brief review of Lieb-Thirring inequalities for eigenvalues of the Schroedinger operator and lower bounds for the quantum mechanical kinetic energy (and some generalizations) in R^n.
In \cite{Nam} Nam proved a Lieb--Thirring Inequality for the kinetic energy of a fermionic quantum system, with almost optimal (semi-classical) constant and a gradient correction term. We present a stronger version of this inequality, with…
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…
We prove analogues of the Lieb-Thirring and Hardy-Lieb-Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed.…
We give a proof of the Lieb-Thirring inequality on the kinetic energy of orthonormal functions by using a microlocal technique, in which the uncertainty and exclusion principles are combined through the use of the Besicovitch covering…
A logarithmic type Lieb-Thirring inequality for two-dimensional Schroedinger operators is established. The result is applied to prove spectral estimates on trapped modes in quantum layers.
We prove on the 2D sphere and on the 2D torus the Lieb-Thirring inequalities with improved constants for orthonormal families of scalar and vector functions.
We prove an analogue of the Lieb--Thirring inequality for many-body quantum systems with the kinetic operator $\sum_i (-\Delta_i)^s$ and the interaction potential of the form $\sum_i \delta_i^{-2s}$ where $\delta_i$ is the nearest-neighbor…
We prove Lieb-Thirring inequalities with improved constants on the two-dimensional sphere and the two-dimensional torus. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative…
Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schr\"odinger operator with a complex-valued potential.
We use a logarithmic Lieb-Thirring inequality for two-dimensional Schroedinger operators and establish estimates on trapped modes in geometrically deformed quantum layers.
In this paper, we obtain new upper bounds for the Lieb-Thirring inequality on the spheres of any dimension greater than $2$. As far as we have checked, our results improve previous results found in the literature for all dimensions greater…
In [8], some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, kinetic equations, and Schr{\"o}dinger type equations with a rotation term. In this work, these exact…
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.
We show that, under very general definitions of a kinetic energy operator $T$, the Lieb-Thirring inequalities for sums of eigenvalues of $T-V$ can be derived from the Sobolev inequality appropriate to that choice of $T$.
This paper investigates two inexact Levenberg-Marquardt (LM) methods for solving systems of nonlinear equations. Both approaches compute approximate search directions by solving the LM linear system inexactly, subject to specific…
We investigate numerically the optimal constants in Lieb-Thirring inequalities by studying the associated maximization problem. We use a monotonic fixed-point algorithm and a finite element discretization to obtain trial potentials which…
We overview some recent results in the field of uncertainty quantification for kinetic equations and related problems with random inputs. Uncertainties may be due to various reasons, such as lack of knowledge on the microscopic interaction…