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 than $2$. We also prove and exhibit an explicit new upper bound for the Lieb-Thirring inequality on $SO(3)$. We also discuss these estimates in the case of general compact Lie groups. Originally developed for estimating the sums of moments of negative eigenvalues of the Schrödinger operator in $L^2(\mathbb{R}^n)$, these inequalities have applications in quantum mechanics and other fields.
This paper is essentially derived from the observation that some results used for improving constants in the Lieb-Thirring inequalities for Schrodinger operators in L2(-\infty,\infty) can be translated to the discrete Schrodinger op- erators and more generally to Jacobi matrices. Some results were previ- ously proved by D. Hundertmark and B. Simon and the aim of this article is to improve the constants obtained in their article [7].
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.
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.
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 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.
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.
We establish Lieb-Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class complex perturbations of periodic and more generally finite gap 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.
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.
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.
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.
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)
For $s\textgreater{}0$, let $H\_0=(-Δ)^s$ be the fractional Laplacian. In this paper, we obtainLieb-Thirring type inequalities for the fractional Schrödinger operator defined as $H=H\_0+V$,where $V \in L^p(\mathbb{R}^d), p\ge 1, d\ge 1,$ is a complex-valued potential.Our methods are based on results of articles by Borichev-Golinskii-Kupin \cite{BoGoKu} and Hansmann \cite{Ha1}.
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.
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.
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.
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.
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.
This paper considers Lieb-Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) - V(x))^{-γ} < C_γ\int_{R^d} V(x)_+^{γ+ d/4} dx for gamma \geq 1 - d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality and where C_γ> 0 is independent of V, is proved for dimensions d = 1,3. As a corollary of this inequality a Sobolev-type inequality is obtained.
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 $T$.
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.
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.
We study to what extent Lieb--Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schrödinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [Complex Anal. Oper. Theory 5, No. 1 (2011), 197-218] and answer another open question raised therein. The results are obtained by means of asymptotic analysis of eigenvalues of discrete Schrödinger operators with rectangular barrier potential and complex coupling. Applying the ideas in the continuous setting, we also solve a similar open problem for one-dimensional Schrödinger …
The sharp one-dimensional Lieb-Thirring inequality for the sum of eigenvalues
Laboratoire de mathématiques d’Orsay, CNRS, Université Paris-Saclay（奥赛数学实验室，法国国家科学研究中心，巴黎萨克雷大学）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学系）
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 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.
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.
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.
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.
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.
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.
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.
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…
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…
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.
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…
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.
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.
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.
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.
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.
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…
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…
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.
We prove bounds of the form $\sum_{e\in I\capσ_\di (H)} \dist (e,σ_\e (H))^{1/2} \leq L^1$-norm of a perturbation, where $I$ is a gap. Included are gaps in continuum one-dimensional periodic Schrödinger operators and finite gap Jacobi matrices where we get a generalized Nevai conjecture about an $L^1$ condition implying a Szegő condition. One key is a general new form of the Birman--Schwinger bound in gaps.
