Inequalities Of Lieb-Thirring Type Research Articles

A simple approach to Lieb–Thirring type inequalities

In [10] 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 a much simplified proof. As a corollary we obtain a simple proof of the original Lieb–Thirring inequality.

A Sharp Lieb-Thirring Inequality for Functional Difference Operators

We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated to mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.

$$L_p$$-Spectrum and Lieb–Thirring Inequalities for Schrödinger Operators on the Hyperbolic Plane

This paper deals with the $L_p$-spectrum of Schr\"odinger operators on the hyperbolic plane. We establish Lieb-Thirring type inequalities for discrete eigenvalues and study their dependence on $p$. Some bounds on individual eigenvalues are derived as well.

Number of Eigenvalues of Non-Self-Adjoint Schrödinger Operators with Dilation Analytic Complex Potentials

In the present paper, we give Lieb—Thirring type inequalities for all isolated eigenvalues of d-dimensional non-self-adjoint Schrodinger operators with complex-valued and dilation analytic potentials. In order to derive them, we prove that isolated eigenvalues and their multiplicities are invariant under complex dilation.

Corrigendum to “Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators” [J. Funct. Anal. 266 (2014) 5016–5044

The short text below corrects a mistake in the proof of [1, Lemma 4.1] .

Note on Lieb-Thirring Type Inequalities for a Complex Perturbation of Fractional Laplacian

Note on Lieb-Thirring Type Inequalities for a Complex Perturbation of Fractional Laplacian

On the distribution of the discrete spectrum of nuclearly perturbed operators in Banach spaces

Let Z 0 be a bounded operator in a Banach space X with purely essential spectrum and K a nuclear operator in X. We construct a holomorphic function the zeros of which coincide with the discrete spectrum of Z 0+K and derive a Lieb-Thirring type inequality. We obtain estimates for the number of eigenvalues in certain regions of the complex plane and an estimate for the asymptotics of the eigenvalues approaching to the essential spectrum of Z 0.

Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators

Let H:=H0+V and H⊥:=H0,⊥+V be respectively perturbations of the unperturbed Schrödinger operators H0 on L2(R3) and H0,⊥ on L2(R2) with constant magnetic field of strength b>0, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H and H⊥. In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.

On Quantitative Bounds on Eigenvalues of a Complex Perturbation of a Dirac Operator

We prove a Lieb-Thirring type inequality for a complex perturbation of a d-dimensional massive Dirac operator $D_m, m\geq 0$ whose spectrum is $]-\infty , -m]\cup[m , +\infty[$. The difficulty of the study is that the unperturbed operator is not bounded from below in this case, and, to overcome it, we use the methods of complex function theory. The methods of the article also give similar results for complex perturbations of the Klein-Gordon operator.

Lieb–Thirring Type Inequalities for Schrödinger Operators with a Complex-Valued Potential

We present a possible generalization of the Lieb–Thirring inequalities for eigenvalues of Schrodinger operators to the case of complex potentials. We ask for a proof or disproof of this generalization. Some weaker results which have been obtained are reviewed.

Applications of inequalities of Lieb-Thirring type to spectral theory

Inequalities of Lieb-Thirring type are established. Applications of these inequalities to estimates of the spectrum of unbounded operators are given.

On inequalities of Lieb-Thirring type

Applying the method proposed by Kashin for proving inequalities of Lieb-Thirring type for orthonormal systems, we prove a similar inequality in the multidimensional case.

On a class of inequalities for orthonormal systems

In this paper, a novel approach to the proof of inequalities of Lieb-Thirring type based on the standard apparatus of the theory of orthogonal series is proposed.

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems

This paper is devoted to inequalities of Lieb–Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues ( λ i ( V ) ) i ∈ N ∗ . We prove that there exists a positive constant C ( γ ) , such that, if γ > d / 2 , then (∗) ∑ i ∈ N ∗ [ λ i ( V ) ] − γ ⩽ C ( γ ) ∫ R d V d 2 − γ d x and determine the optimal value of C ( γ ) . Such an inequality is interesting for studying the stability of mixed states with occupation numbers. We show how the infimum of λ 1 ( V ) γ ⋅ ∫ R d V d 2 − γ d x on all possible potentials V, which is a lower bound for [ C ( γ ) ] −1 , corresponds to the optimal constant of a subfamily of Gagliardo–Nirenberg inequalities. This explains how (∗) is related to the usual Lieb–Thirring inequality and why all Lieb–Thirring type inequalities can be seen as generalizations of the Gagliardo–Nirenberg inequalities for systems of functions with occupation numbers taken into account. We also state a more general inequality of Lieb–Thirring type (∗∗) ∑ i ∈ N ∗ F ( λ i ( V ) ) = Tr [ F ( − Δ + V ) ] ⩽ ∫ R d G ( V ( x ) ) d x , where F and G are appropriately related. As a special case corresponding to F ( s ) = e − s , (∗∗) is equivalent to an optimal Euclidean logarithmic Sobolev inequality ∫ R d ρ log ρ d x + d 2 log ( 4 π ) ∫ R d ρ d x ⩽ ∑ i ∈ N ∗ ν i log ν i + ∑ i ∈ N ∗ ν i ∫ R d | ∇ ψ i | 2 d x , where ρ = ∑ i ∈ N ∗ ν i | ψ i | 2 , ( ν i ) i ∈ N ∗ is any nonnegative sequence of occupation numbers and ( ψ i ) i ∈ N ∗ is any sequence of orthonormal L 2 ( R d ) functions.

Lieb-Thirring integral inequalities and their applications to attractors of the Navier-Stokes equations

Integral inequalities of Lieb-Thirring type and their generalizations are proved. All the corresponding constants are given in explicit form. Special attention is devoted to applications to the attractors of the two-dimensional Navier-Stokes equations. In particular, an explicit two-sided estimate of the attractor dimension is established for the Kolmogorov problem on the two-dimensional elongated torus.

Uniform Lieb-Thirring Inequality for the Three-Dimensional Pauli Operator with a Strong Non-Homogeneous Magnetic Field

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. A new Lieb-Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.