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Abstract We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is convex, we obtain universal, non-asymptotic bounds that correctly reproduce the two leading terms in the asymptotics and depend on the domain only through simple geometric characteristics. Important ingredients in our proof are non-asymptotic versions of various Tauberian theorems. 

Lieb and Carlen have shown that mixed states with minimal Wehrl entropy are coherent states. We prove that mixed states with almost minimal Wehrl entropy are almost coherent states. This is proved in a quantitative sense where both the norm and the exponent are optimal and the constant is explicit. We prove a similar bound for generalized Wehrl entropies. As an application, a sharp quantitative form of the log-Sobolev inequality for functions in the Fock space is provided. 

Abstract We consider the problem of minimizing the lowest eigenvalue of the Schrödinger operator −Δ +Vin $L^2\left({\mathbb{R}}^d\right)$$${L}^{2}({\mathbb{R}}^{d})$$when the integral ∫e^−tV dxis given for somet> 0. We show that the eigenvalue is minimal for the harmonic oscillator and derive a quantitative version of the corresponding inequality. 

abstract: The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $$N$$ lowest eigenvalues of a Schr\odinger operator $$-\Delta-V(x)$ in terms of an $$L^p(\mathbb{R}^d)$$ norm of the potential $$V$$. We prove here the existence of an optimizing potential for each $$N$$, discuss its qualitative properties and the Euler--Lagrange equation (which is a system of coupled nonlinear Schr\odinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition $$\gamma>\max\{0,2-d/2\}$ on the Riesz exponent in the inequality, we prove the compactness of all the optimizing sequences up to translations. We also show that the optimal Lieb-Thirring constant cannot be stationary in $$N$$, which sheds a new light on a conjecture of Lieb-Thirring. In dimension $d=1$ at $$\gamma=3/2$$, we show that the optimizers with $$N$$ negative eigenvalues are exactly the Korteweg-de Vries $$N$$-solitons and that optimizing sequences must approach the corresponding manifold. Our work also covers the critical case $$\gamma=0$$ in dimension $$d\geq3$$ (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem. 

We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel’s theorem, which states that, as soon as the dimension is strictly larger than $1$, the number of nodal domains of an eigenfunction corresponding to the $k$-th eigenvalue is strictly (and uniformly, in a certain sense) smaller than $k$for large  $k$. In the first part of this paper we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups. In the second part, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way, we improve known bounds on the optimal constants in the Faber–Krahn and isoperimetric inequalities on these groups. 

Получена двучленная спектральная асимптотика для средних Риссасобственных значений оператора Лапласа на липшицевой области с граничными условиями Робена.Второе слагаемое в асимптотике оказывается тем же самым,что и в случае граничных условий Неймана. Такая асимптотика установленадля средних Рисса произвольного положительного порядка.Для порядков один и выше и при дополнительных предположениях о функции,входящей в граничные условия, также найден старший член асимптотикидля разности между средними Рисса собственных значений задач Робена и Неймана.