The Lieb-Thirring inequalities: Recent results and open problems

Lieb--Thirring bounds for complex Jacobi matrices

We obtain various versions of classical Lieb--Thirring bounds for one- and multi-dimensional complex Jacobi matrices. Our method is based on Fan-Mirski Lemma and seems to be fairly general.

Best constants in Lieb-Thirring inequalities: a numerical investigation

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 provide lower bounds on the optimal constants. We examine the one-dimensional and radial cases in detail. Our numerical results provide new lower bounds, insight into the behavior of the maximizers and confirm some existing conjectures. Based on our numerical results, we formulate a complete conjecture about the best constants for all possible values of the pa…

Fractional Hardy-Lieb-Thirring and related inequalities for interacting systems

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. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.

Lieb-Thirring estimates for singular measures

Lieb-Thirring type estimates are proved for the sum of powers of negative eigenvalues of a Schrödinger type operator $(-Δ)^l -Vμ$ where $μ$ is a singular measure in $\mathbb{R}^d,$ satisfying a condition on the measure of balls and $V$ is a $μ$-measurable function.

$L_p$-spectrum and Lieb-Thirring inequalities for Schrödinger operators on the hyperbolic plane

This paper deals with the $L_p$-spectrum of Schrödinger 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.

Sharp Lieb-Thirring Inequalities in High Dimensions

We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schrödinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{γ,d}$ with $γ\ge 3/2$ and arbitrary $d\ge 1$. (revised, to appear in Acta Math)

Lieb-Thirring Inequalities for Finite and Infinite Gap Jacobi Matrices

We establish Lieb-Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class perturbations under very general assumptions. Our results apply, in particular, to perturbations of reflectionless Jacobi operators with finite gap and Cantor-type essential spectrum.

The state of the Lieb--Thirring conjecture

In 1976 Lieb and Thirring established upper bounds on sums of powers of the negative eigenvalues of a Schrödinger operator in terms of semiclassical phase-space integrals. Over the last 45 years the optimal constants in these inequalities, the values of which were conjectured by Lieb and Thirring, have been subject of intense investigations. We aim to review existing results.

Lieb-Thirring and Cwickel-Lieb-Rozenblum inequalities for perturbed graphene with a Coulomb impurity

We study the two dimensional massless Coulomb-Dirac operator restricted to its positive spectral subspace and prove estimates on the negative eigenvalues created by electromagnetic perturbations.

Equivalence of Sobolev inequalities and Lieb-Thirring inequalities

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 $T$.

带复势的薛定谔算子与雅可比算子的最优Lieb-Thirring型不等式

Optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials

该研究针对带复势的薛定谔与雅可比算子，证明了含距离权重的最优Lieb-Thirring型不等式，建立非可积权重的发散估计，得到比实势半经典方法更优的对数或多项式级发散速率。

The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities

In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb-Thirring constant when the eigenvalues of a Schrödinger operator $-Δ+V(x)$ are raised to the power $κ$ is never given by the one-bound state case when $κ>\max(0,2-d/2)$ in space dimension $d\geq1$. When in addition $κ\geq1$ we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo…

Littlewood-Paley decomposition of operator densities and application to a new proof of the Lieb-Thirring inequality

The goal of this note is to prove a analogue of the Littewood-Paley decomposition for densities of operators and to use it in the context of Lieb-Thirring inequalities.

Critical Lieb-Thirring bounds for one-dimensional Schrodinger operators and Jacobi matrices with regular ground states

This paper has been withdrawn by the author in favor of a stronger result proven by the author with R. Frank and T. Weidl in arXiv:0707.0998

Lieb-Thirring inequalities for wave functions vanishing on the diagonal set

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 of the configuration space, without any statistical assumption on the particles.

Lieb-Thirring inequalities for an effective Hamiltonian of bilayer graphene

Combining the methods of Cuenin [2019] and Borichev-Golinskii-Kupin [2009, 2018], we obtain the so-called Lieb-Thirring inequalities for non-selfadjoint perturbations of an effective Hamiltonian for bilayer graphene.

A simple proof of Hardy-Lieb-Thirring inequalities

We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schroedinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Soerensen, and Spitzer. Moreover, we prove that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a larger constant).

带复势的随机薛定谔算子的Lieb-Thirring型不等式

Lieb-Thirring-type inequalities for random Schrödinger operators with complex potentials

该研究综述带复势随机薛定谔算子特征值界的相关成果与证明，提出预解式的新Schatten范数估计方法，以此推导特征值和的界。

The periodic Lieb-Thirring inequality

We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D Lieb-Thirring inequality at the special exponent $γ=3/2$ admits a one-parameter family of periodic optimizers, interpolating between the one-bound state and the uniform potential. Finally, we provide numerical simulations in 2D which support our conjecture that optimizers could be periodic.

关于带复势的多维薛定谔算子的Lieb-Thirring不等式

On Lieb-Thirring inequalities for multidimensional Schrödinger operators with complex potentials

本研究通过构造反例，解决了带复势的多维薛定谔算子的Lieb-Thirring不等式推广这一公开问题，将已有的一维反例结果拓展到了更高维度。

From Lieb-Thirring inequalities to spectral enclosures for the damped wave equation

Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the one-dimensional damped wave operator is similar to the undamped one provided that the L^1 norm of the (possibly complex-valued) damping is less than 2. It follows that these small dampings are spectrally undetectable.

Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials

Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential.

Lieb-Thirring estimates for non self-adjoint Schrödinger operators

For general non-symmetric operators $A$, we prove that the moment of order $γ\ge 1$ of negative real-parts of its eigenvalues is bounded by the moment of order $γ$ of negative eigenvalues of its symmetric part $H = {1/2} [A + A^*].$ As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schrödinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment of resonances of Schrödinger self-adjoint operators.

Direct methods to Lieb-Thirring kinetic inequalities

We review some recent progress on Lieb-Thirring inequalities, focusing on direct methods to kinetic estimates for orthonormal functions and applications for many-body quantum systems.

Spectral inequalities for Jacobi operators and related sharp Lieb-Thirring inequalities on the continuum

In this paper we approximate a Schrödinger operator on $L^2(\R)$ by Jacobi operators on $\ell^2(\Z)$ to provide new proofs of sharp Lieb-Thirring inequalities for the powers $γ=1/2$ and $γ=3/2$. To this end we first investigate spectral inequalities for Jacobi operators. Using the commutation method we present a new, direct proof of a sharp inequality corresponding to a Lieb-Thirring inequality for the power 3/2 on $\ell^2(\Z)$. We also introduce inequalities for higher powers of the eigenvalues as well as for matrix-valued potentials and compare our results to previously established bounds.

Lieb-Thirring inequality with semiclassical constant and gradient error term

In 1975, Lieb and Thirring derived a semiclassical lower bound on the kinetic energy for fermions, which agrees with the Thomas-Fermi approximation up to a constant factor. Whenever the optimal constant in their bound coincides with the semiclassical one is a long-standing open question. We prove an improved bound with the semiclassical constant and a gradient error term which is of lower order.

Lieb-Thirring Bound for Schrödinger Operators with Bernstein Functions of the Laplacian

A Lieb-Thirring bound for Schrödinger operators with Bernstein functions of the Laplacian is shown by functional integration techniques. Several specific cases are discussed in detail.

二维Pauli算子的Lieb-Thirring不等式

Lieb-Thirring inequality for the 2D Pauli operator

该研究针对二维Pauli算子，在归一化磁通量|α|<1的零能共振情形下，证明了γ次矩的Lieb-Thirring不等式在γ≥|α|且γ>0的最优约束下成立，改进了此前仅适用于γ≥1阶矩的结果。

Lieb-Thirring inequalities for higher order differential operators

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order gamma >= 3/4 for fourth order Schrödinger operators in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators. For the critical case gamma = 1 - 1/2l in dimension d=1 with differential order 2l >= 4 we prove the strict inequality L^0(l,gamma,d) < L(l,gamma,d), which holds in contrast to current conjectures.

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

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 sharp constant for these exponents should be attained by potentials having only one bound state. Here we exhibit a connection between the Lieb--Thirring conjecture for gamma=1 and an isporimetric inequality for ovals in the plane.

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.

Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics

In one and two spatial dimensions there is a logical possibility for identical quantum particles different from bosons and fermions, obeying intermediate or fractional (anyon) statistics. We consider applications of a recent Lieb-Thirring inequality for anyons in two dimensions, and derive new Lieb-Thirring inequalities for intermediate statistics in one dimension with implications for models of Lieb-Liniger and Calogero-Sutherland type. These inequalities follow from a local form of the exclusion principle valid for such generalized exchange statistics.

Lieb-Thirring inequalities on the half-line with critical exponent

We consider a Schrödinger operator on the half-line with a Dirichlet boundary condition at the origin and show that moments of its negative eigenvalues can be estimated by the part of the potential that is larger than the critical Hardy weight. The estimate is valid for the critical value of the moment parameter.

半直线矩阵薛定谔算子的反向Lieb-Thirring不等式

Reverse Lieb-Thirring inequality for the half-line matrix Schrödinger operator

本文针对半直线上的矩阵薛定谔方程，证明了具有精确常数的反向Lieb-Thirring不等式。

重探Lieb-Thirring不等式

The Lieb-Thirring inequality revisited

该研究针对薛定谔算子负特征值之和的Lieb-Thirring不等式最优常数问题，提出新的估计方法，显著提升了现有界的精度。

Universal Monotonicity of Eigenvalue Moments and Sharp Lieb-Thirring Inequalities

We show that phase space bounds on the eigenvalues of Schrödinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In particular, we provide a new proof of sharp Lieb-Thirring inequalities.

Another Application of Dilation Analytic Method for Complex Lieb--Thirring Type Estimates

We consider non-self-adjoint Schrödinger operators $H_{\rm c}=-Δ+V_{\rm c}$ (resp. $H_{\rm r}=-Δ+V_{\rm r}$) acting in $L^2(\mathbb R^d)$, $d\ge 1$, with dilation analytic complex (resp. real) potentials. We were able to find out perhaps a new application of dilation analytic method in \cite{So1} (N. Someyama, "Number of Eigenvalues of Non-self-adjoint Schrödinger Operators with Dilation Analytic Complex Potentials," Reports on Mathematical Physics, Volume 83, Issue 2, pp.163-174 (2019).). We give a Lieb--Thirring type estimate on resonance eigenvalues of $H_{\rm c}$ in the open complex sector…

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

We prove a Lieb-Thirring type inequality for potentials such that the associated Schrödinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg inequality for systems. As a special case, we prove a logarithmic Sobolev inequality for infinite systems of mixed states. Optimal constants are determined and free energy estimates in connection with mixed states representations are also investigated.

具有逆最近邻相互作用的大量子系统的Lieb--Thirring不等式

Lieb--Thirring inequalities for large quantum systems with inverse nearest-neighbor interactions

本文针对动能算子为分数阶拉普拉斯算子、相互作用势为逆最近邻距离幂次的多体量子系统，证明了无需费米子反对称假设的Lieb--Thirring型不等式，并进一步给出Hardy--Lieb--Thirring型结果及强耦合极限下最优常数的渐近行为。

有限秩Lieb-Thirring不等式的最优化问题

Optimizers for the finite-rank Lieb-Thirring inequality

本文证明有限秩Lieb-Thirring不等式存在优化势，研究其性质与Euler-Lagrange方程，证明优化序列紧性，并揭示最优常数与孤子及Yamabe问题不变量的联系。

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

Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively perturbations of the free Schrödinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ and $H_{0,\perp}$ on $L^{2}\big(\mathbb{R}^{2d}\big)$, $d \geq 1$ with constant magnetic field of strength $b>0$, and $V$ is a complex relatively compact perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of $H$ and $H_{\perp}$. In particular, these estimates give $a\, priori$ information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast se…

Lieb--Thirring型不等式的一个简单方法

A simple approach to Lieb--Thirring type inequalities

本文提出了Nam所证明的费米子量子系统Lieb--Thirring不等式的一个更强版本，通过大幅简化的证明方法，并作为推论给出了原始Lieb--Thirring不等式的简单证明。

A proof of the Lieb-Thirring inequality via the Besicovitch covering lemma

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 lemma.

Lieb-Thirring and Jensen sums for non-self-adjoint Schrödinger operators on the half-line

We prove upper and lower bounds for sums of eigenvalues of Lieb-Thirring type for non-self-adjoint Schrödinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases.

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.

Sum Rules for Jacobi Matrices and Divergent Lieb-Thirring Sums

Extending earlier work of Killip-Simon and Simon-Zlatos, we obtain sum rules for Jacobi matrices in which the a.c. part of the spectral measure and the eigenvalues of the matrix appear on opposite sides of the equation. We use these to obtain various results on divergence of certain Lieb-Thirring sums of eigenvalues and Szego-type integrals of the a.c. part of the spectral measure.

Lieb-Thirring type bounds for perturbed Schrödinger operators with single-well potentials

We prove an upper bound on the sum of the distances between the eigenvalues of a perturbed Schrödinger operator $H_0-V$ and the lowest eigenvalue of $H_0$. Our results hold for operators $H_0=-Δ-V_0$ in one dimension with single-well potentials. We rely on a variation of the well-known commutation method. In the Pöschl-Teller and Coulomb cases we are able to use the explicit factorisations to establish improved bounds.

Lieb-Thirring inequality on the four-dimensional sphere and torus

In this paper, we mainly study the Lieb-Thirring inequality for families of orthonormal scalar functions on the four-dimensional sphere $ \mathbb{S}^{4} $ and torus $ \mathbb{T}^{4} $. The bounds of all the constants involved are obtained. Specifically, we prove that the costant $ \mathcal{K}_{4}(\mathbb{S}^{4}) $ of the Lieb-Thirring inequality on the sphere $ \mathbb{S}^{4} $ satisfies $$ 0.0844 \leq \mathcal{K}_{4}(\mathbb{S}^{4})\leq 0.1728,$$ and the constant $ \mathcal{K}_{4}(\mathbb{T}^{4}) $ of the Lieb-Thirring inequality on the torus $ \mathbb{T}^{4} $ satisfies $$ 0.0190 \leq \mathc…

Hardy-Lieb-Thirring inequalities for fractional Schroedinger operators

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schroedinger-like operators remain true, with possibly different constants, when the critical Hardy-weight $C|x|^{-2}$ is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schroedinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of r…

A positive density analogue of the Lieb-Thirring inequality

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schrödinger 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 of orthonormal functions. Here we extend these to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials.

A Lieb-Thirring inequality for extended anyons

We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. That is, we consider 2D fermionic particles coupled to magnetic flux tubes of non-zero radius, and prove a Lieb-Thirring inequality for the corresponding many-body kinetic energy operator. The implied constant is independent of the radius of the flux tubes, and proportional to the statistics parameter.

Lieb-Thirring Inequality for a Model of Particles with Point Interactions

We consider a model of quantum-mechanical particles interacting via point interactions of infinite scattering length. In the case of fermions we prove a Lieb-Thirring inequality for the energy, i.e., we show that the energy is bounded from below by a constant times the integral of the particle density to the power 5/3.

Lieb-Thirring Bounds for Interacting Bose Gases

We study interacting Bose gases and prove lower bounds for the kinetic plus interaction energy of a many-body wave function in terms of its particle density. These general estimates are then applied to various types of interactions, including hard sphere (in 3D) and hard disk (in 2D) as well as a general class of homogeneous potentials.

Hardy and Lieb-Thirring inequalities for anyons

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter $α\in [0,1]$ ranging from bosons ($α=0$) to fermions ($α=1$). We prove a (magnetic) Hardy inequality for anyons, which in the case that $α$ is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by Dyson and Lenard's original approach to the stability of fermionic matter in three dimensions, we prove a Lieb-Thirring inequality for these types of anyons.