Alexei Ilyin, Ari Laptev, Sergey Zelik
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 eigenvalues.
We consider the Lieb-Thirring inequalities on the d-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 the $γ$-moments of the negative eigenvalues with constants independent of ratio of the periods. Applications to the attractors of the damped Navier-Stokes system are given.
Magnetic Lieb--Thirring inequalities on the torus
该研究针对环面上的磁性 Schrödinger 算子建立 Lieb–Thirring 不等式，并指出相关常数由磁通量决定。
We prove on the sphere $\mathbb{S}^2$ the Lieb--Thirring inequalities for orthonormal families of scalar and vector functions both on the whole sphere and on proper domains on $\mathbb{S}^2$. By way of applications we obtain an explicit estimate for the dimension of the attractor of the Navier--Stokes system on a domain on the sphere with Dirichlet non-slip boundary conditions.
The sharp one-dimensional Lieb-Thirring inequality for the sum of eigenvalues
本文证明一维Lieb-Thirring不等式（指数为1）的最佳常数为$4/(3\sqrt{3}\pi)$，证实了该情形下的Lieb-Thirring猜想。
We attempt to survey recent results and open problems connected to Lieb-Thirring inequalities.
We discuss the estimates for the $L^p$-norms of systems of functions that are orthonormal in $L^2$ and $H^1$, respectively, and their essential role in deriving good or even optimal bounds for the dimension of global attractors for the classical Navier--Stokes equations and for a class of $α$-models approximating them. New applications to interpolation inequalities on the 2D torus are also given.
We consider an analogue of the Lieb-Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb-Thirring constant converges to the optimal constant of the one-body Gagliardo-Nirenberg interpolation inequality without interaction.
Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schroedinger operators.
Improved estimates on the constants $L_{γ,d}$, for $1/2<γ<3/2$, $d\in N$ in the inequalities for the eigenvalue moments of Schrödinger operators are established.
We improve the Lieb-Thirring type inequalities by Demuth, Hansmann and Katriel (J. Funct. Anal. 2009) for Schrödinger operators with complex-valued potentials. Our result involves a positive, integrable function. We show that in the one-dimensional case the result is sharp in the sense that if we take a non-integrable function, then an analogous inequality cannot hold.
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.
This note is devoted to Keller-Lieb-Thirring spectral estimates for Schrödinger operators on infinite cylinders: the absolute value of the ground state level is bounded by a function of a norm of the potential. Optimal potentials with small norms are shown to depend on a single variable. The proof is a perturbation argument based on recent rigidity results for nonlinear elliptic equations on cylinders. Conversely, optimal single variable potentials with large norms must be unstable. The optimal threshold between the two regimes is established in the case of the product of a sphere by a line.
Lieb--Thirring inequalities on manifolds with constant negative curvature
本文研究负常曲率流形上的 Lieb–Thirring 不等式，利用双曲空间连续谱阈值证明谱估计，并导出非尖锐常数的 Pólya 型不等式及二维数值例证。
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…
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.
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。
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。
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…
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。
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 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。
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.
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。
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 under consideration are either related to Sobolev type inequalities or to Fourier restriction type inequalities.
Keller and Lieb-Thirring estimates of the eigenvalues in the gap of Dirac operators
该研究针对带质量Dirac算子本质谱能隙的最低本征值，借助势的Lebesgue范数建立了对应Schrödinger算子情形的Keller型估计，给出了临界界与最优势，并推广为能隙本征值的Lieb-Thirring不等式，结果多基于Birman-Schwinger重述得到。
Optimizers for the finite-rank fractional Lieb--Thirring inequality
该研究针对有限秩分数阶Lieb-Thirring不等式，证明其优化器存在性，刻画优化谱性质，推导归一化最大化序列预紧性与最优常数绑定不等式，明确最优常数无法由有限负特征值势达到。
On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities
本研究通过函数傅里叶变换$L^1$范数的变分刻画、对偶性及复平面带状区域Hardy型全纯函数空间分析，显式求解了与CLR和LT不等式最优常数上界相关的变分问题，得到对应方法下的最佳上界。
We derive spectral estimates of the Lieb-Thirring type for eigenvalues of Dirichlet Laplacians on strictly shrinking spiral-shaped domains.
Optimizers of the Finite-Rank Hardy-Lieb-Thirring Inequality for Hardy-Schrödinger Operator
本文研究Hardy-Schrödinger算子有限秩Hardy-Lieb-Thirring不等式的优化子，分析其存在性与解析性质。
On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems
研究伪相对论费米系统在$L^2$临界情况下的基态存在性及极限行为，证明基态存在当且仅当吸引力强度$a$满足$0 < a < D_{4/3,2}$，并探讨$a$趋近于$D_{4/3,2}$时基态的行为。
We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains Ω\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators on domains.
In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain Lieb-Thirring inequalities for magnetic Schrodinger operators on multi-dimensional cylinders.
We prove Lieb-Thirring-type bounds for fractional Schrödinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of Frank and Sabin.
Ground States of Attractive Fermi Schrödinger Systems with Ring-Shaped Potentials
本文研究了在环形势能中被束缚的费米-薛定谔系统基态的存在性与非存在性，以及临界参数下的质量集中行为。
A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in $\mathbb{R}^d$ has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of $\mathbb{R}^d$, in particular for cubes, has arised. The purpose of this manuscript is two-fold. First we prove a GNS inequality for convex domains, with explicit constants which depend on the geometry of the domain. Later, using the discrete version of Rumin's method, we prove GNS inequalities on cubes with improved constants.
This paper is devoted to the Lin-Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below an explicit bound that we relate with an optimal constant for a Gagliardo-Nirenberg-Sobolev interpolation inequality and also with an optimal Keller-Lieb-Thirring inequality. Our results are valid in a sub-linear regime as well. The rigidity bound is obtained by nonlinear flow meth…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。