Related papers: Lieb-Thirring constant on the sphere and on the to…

We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{\gamma,d}$ with $\gamma\ge 3/2$ and arbitrary $d\ge…

In this paper, motivated by recent important works due to Frank-Lewin-Lieb-Seiringer \cite{FLLS} and Frank-Sabin \cite{frank-sabin-1}, we study the Strichartz inequality on torus with the orthonormal system input and obtain sharp estimates…

Improved estimates on the constants $L_{\gamma,d}$, for $1/2<\gamma<3/2$, $d\in N$ in the inequalities for the eigenvalue moments of Schr\"{o}dinger operators are established.

In this short note we prove Lieb--Thirring inequalities on manifolds with negative constant curvature. The discrete spectrum appears below the continuous spectrum $(d-1)^2/4, \infty)$, where $d$ is the dimension of the hyperbolic space. As…

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 a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as…

We prove stable versions of trace theorems on the sphere in $L^2$ with optimal constants, thus obtaining rather precise information regarding near-extremisers. We also obtain stability for the trace theorem into $L^q$ for $q > 2$, by…

We prove that, given a planar bi-Lipschitz homeomorphism $u$ defined on the boundary of the unit square, it is possible to extend it to a function $v$ of the whole square, in such a way that $v$ is still bi-Lipschitz. In particular,…

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schr\"odinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider,…

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…

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…

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 revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schr\"odinger equation in 2D on a square torus, and we prove the equivalent result on any tori.

In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue $\lambda_2(T,g)$ on a torus $T$, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for…

In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the…

Using numerical simulations, the stability and scattering properties of the O(3) model on a two-dimensional torus are studied. Its solitons are found to be unstable but can be stabilized by the addition of a Skyrme term to the Lagrangian.…

In this paper we show a new inequality which generalizes to the unit sphere the Lebedev-Milin inequality of the exponentiation of functions on the unit circle. It may also be regarded as the counterpart on the sphere of the second…

We present a new viewpoint (namely, reproducing kernels) and new proofs for several recent results of J. Geronimo and H. Woerdeman on orthogonal polynomials on the two dimenional torus (and related subjects). In addition, we show how their…

We prove that the optimal constant in the Lieb--Thirring inequality on a star graph with $N$ edges coincides with that on $\mathbb R$ if $N$ is even. For odd $N$ we show that this property holds when restricting to radial potentials and we…

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R}…