We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin-Li-Yau and Kr & ouml,.ger, valid for Riesz exponents , extend to certain values , provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of P & oacute,.lya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.

In this paper we characterise the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic. In particular we show that, if the Fourier transform of the repulsive potential is positive, the minimiser is supported on a fully-dimensional ellipsoid, and its density is given by a Barenblatt-type profile. Our technique of proof is based on a Fourier representation of the potential of such measures, that extends a previous formula established by some of the authors in the Coulomb case.

We consider the problem of minimizing the lowest eigenvalue of the Schrodinger operator -Delta + V in L-2(Rd) when the integral integral e (-tV ) dx is given for some t > 0. We show that the eigenvalue is minimal for the harmonic oscillator and derive a quantitative version of the corresponding inequality.

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.

We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as in the case of Neumann boundary conditions. This is valid for Riesz means of arbitrary positive order. For orders at least one and under additional assumptions on the function determining the boundary conditions, we derive leading order asymptotics for the difference between Riesz means of Robin and Neumann eigenvalues.

We prove a matrix inequality for convex functions of a Hermitian matrix on a bipartite space. As an application, we reprove and extend some theorems about eigenvalue asymptotics of Schr & ouml,.dinger operators with homogeneous potentials. The case of main interest is where the Weyl expression is infinite and a partially semiclassical limit occurs.

The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the N lowest eigenvalues of a Schrodinger operator -triangle- V(x) in terms of an L-p(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 Schro<spacing diaeresis>dinger 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 >= 3 (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem.

In this paper, we study the sharp constants in fractional Sobolev inequalities associated with the regional fractional Laplacian in domains.

We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that in an optimal part of the remaining parameter regime all minimizers are uniform distributions on a surface of a sphere, thus showing concentration on a lower dimensional set. Our method of proof uses convexity estimates on hypergeometric functions.

We study the Pauli operator in a 2-dimensional, connected domain with Neumann or Robin boundary condition. We prove a sharp lower bound on the number of negative eigenvalues reminiscent of the Aharonov-Casher formula. We apply this lower bound to obtain a new formula on the number of eigenvalues of the magnetic Neumann Laplacian in the semiclassical limit. Our approach relies on reduction to a boundary Dirac operator. We analyze this boundary operator in two different ways. The first approach uses Atiyah-Patodi-Singer (APS) index theory. The second approach relies on a conservation law for the Benjamin-Ono equation.

By the Aharonov-Casher theorem, the Pauli operator P has no zero eigenvalue when the normalized magnetic flux alpha satisfies |alpha| < 1, but it does have a zero energy resonance. We prove that in this case a Lieb-Thirring inequality for the gamma-th moment of the eigenvalues of P + V is valid under the optimal restrictions gamma >= |alpha| and gamma > 0. Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order gamma >= 1.

- 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.

We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin-Talenti functions.

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.

The Berezin-Li-Yau and the Kroger inequalities show that Riesz means of order >= 1 of the eigenvalues of the Laplacian on a domain Q of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a ' multiplicative factor that depends only on the dimension and the product root Lambda|Q|(1/d), where A is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when |Q|( 1/d) is replaced by a generalized inradius of Omega. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.

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.

The paper is devoted to a family of discrete one-dimensional Schodingeroperators with power-like decaying potentials with rapid oscillations. In particular, the potential V(n)=lambda n(-alpha)cos(pi omega n beta)with1<beta<2 alpha, it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian

Given 0<s<(d)/(2 )with s <= 1, we are interested in the large N-behavior of the optimal constant kappa N in the Hardy inequality ∑(N)(n=1)(-Delta(n))(s)>=kappa(N)∑(n<m)|X-n-X-m|(-2s), when restricted to antisymmetric functions. We show that N1-2s/d kappa(N) has a positive, finite limit given by a certain variational problem, thereby generalizing a result of Lieb and Yau related to the Chandrasekhar theory of gravitational collapse.

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of the Hardy-Schr & ouml,.dinger operator -Delta-(d-2)(2)/(4|x|(2))-W(x) on L-2(R-d). The bound is given in terms of a weighted Ld/2-norm of W which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian. (c) 2024 The Author(s). Published by Elsevior Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We study the trace ideal properties of the commutators [( - Delta) is an element of/2 , M- f ] of a power of the Laplacian with the multiplication operator by a function f on R- d . For a certain range of is an element of is an element of R, we show that this commutator belongs to the weak L Schatten class L d/ 1 - is an element of , infinity if and only if the distributional gradient of f belongs to L d/1-is an element of . Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC license (http:// creativecommons .org /licenses /by -nc /4 .0/).

It is shown that parallel to A parallel to(2)(Ld)>= d/d-2S(d )is a necessary condition for the existence of a nontrivial solution psi of the Dirac equation gamma & sdot,.(-i del-A)psi=0 in d dimensions. Here, S-d is the sharp Sobolev constant. If d is odd and parallel to A parallel to(2)(Ld)=d/d-2S(d), then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.

We show that the Caffarelli-Kohn-Nirenberg (CKN) inequality holds with a remainder term that is quartic in the distance to the set of optimizers for the full parameter range of the Felli-Schneider (FS) curve. The fourth power is best possible. This is due to the presence of non-trivial zero modes of the Hessian of the deficit functional along the FS-curve. Following an iterated Bianchi-Egnell strategy, the heart of our proof is verifying a 'secondary non-degeneracy condition'. Our result completes the stability analysis for the CKN-inequality to leading order started by Wei and Wu. Moreover, it is the first instance of degenerate stability for non-constant optimizers and for a non-compact domain.

. We derive weighted versions of the Cwikel-Lieb-Rozenblum inequality for the Schro center dot dinger operator in two dimensions with a nontrivial Aharonov-Bohm magnetic field. Our bounds capture the optimal dependence on the flux and we identify a class of long-range potentials that saturate our bounds in the strong coupling limit. We also extend our analysis to the twodimensional Schro center dot dinger operator acting on antisymmetric functions and obtain similar results.

We consider Hardy operators on the half-space, that is, ordinary and fractional Schrodinger operators with potentials given by the appropriate power of the distance to the boundary. We show that the scales of homogeneous Sobolev spaces generated by the Hardy operators and by the fractional Laplacian are comparable with each other when the coupling constant is not too large in a quantitative sense. Our results extend those in the whole Euclidean space and rely on recent heat kernel bounds.

We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Holder inequalities for polynomials, conjectured by Bodmann.

We review some older and more recent results concerning the energy and particle distribution in ground states of heavy Coulomb systems. The reviewed results are asymptotic in nature: they describe properties of many-particle systems in the limit of a large number of particles. Particular emphasis is put on models that take relativistic kinematics into account. While non-relativistic models are typically rather well understood, this is generally not the case for relativistic ones and leads to a variety of open questions.

We obtain Weyl type asymptotics for the quantised derivative d over bar f of a function f from the homgeneous Sobolev space Wd1 (Rd) on Rd. The asymptotic coefficient II backward difference fIILd(Rd) is equivalent to the norm of d over bar f in the principal ideal Ld,infinity, thus, providing a non-asymptotic, uniform bound on the spectrum of d over bar f. Our methods are based on the C*-algebraic notion of the principal symbol mapping on Rd, as developed recently by the last two authors and collaborators.

We provide a brief, but self-contained, introduction to the theory of self-adjoint operators. In a first section we give the relevant definitions, including that of the spectrum of a self-adjoint operator, and we discuss the proof of the spectral theorem. In a second section, we discuss the connection between lower semibounded, self-adjoint operators and lower semibounded, closed quadratic forms, and we derive the variational characterization of eigenvalues in the form of Glazman’s lemma and of the Courant–Fischer–Weyl min-max principle. Furthermore, we discuss continuity properties of Riesz means and present in abstract form the Birman–Schwinger principle.

./ Abstract. We show that on S1(1= d -2) x Sd-1(1) the conformally invariant Sobolev inequal-ity holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is moti-vated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on Sd. Our proof proceeds by an iterated Bianchi-Egnell strategy.

We consider Lane-Emden ground states with polytropic index 0 <= q - 1 <= 1, that is, minimizers of the Dirichlet integral among L-q-normalized functions. Our main result is a sharp lower bound on the L-2-norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets Omega subset of R-d, without assuming convexity.

This paper presents some results concerning the size of magnetic fields that support zero modes for the three-dimensional Dirac equation and related problems for spinor equations. It is a well-known fact that for the Schrodinger equation in three dimensions to have a negative energy bound state, the 3/2 norm of the potential has to be greater than the Sobolev constant. We prove an analogous result for the existence of zero modes, namely that the 3/2 norm of the magnetic field has to greater than twice the Sobolev constant. The novel point here is that the spinorial nature of the wave function is crucial. It leads to an improved diamagnetic inequality from which the bound is derived. While the results are probably not sharp, other equations are analyzed where the results are indeed optimal.

We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions N >= 6. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions N >= 3, for homogeneous interaction potentials with higher power.

We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modeling of aggregation phenomena. We show that in a certain regime minimizers are absolutely continuous with an unbounded density, thereby settling a question that was left open in previous works. (c) 2021 Elsevier Ltd. All rights reserved.

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the n-sphere involving an operator of order 2s > n. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of sand settle the question of validity of a corresponding inequality in all dimensions n >= 2. (c) 2021 Elsevier Inc. All rights reserved.

We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

Our recent work on the Burchard-Choksi-Topaloglu flocking problem showed that in the large mass regime the ground state density profile is the characteristic function of some set. Here we show that this set is, in fact, a round ball. The essential mathematical structure needed in our proof is a strict rearrangement inequality with a quantitative error estimate, which we deduce from recent deep results of M. Christ.

In dimension two, we investigate a free energy and the ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrodinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy-Littlewood-Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

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 Schrodinger operator -Delta + V(x) are raised to the power. is never given by the one-bound state case when kappa > max(0, 2 - d/2) in space dimension d >= 1. When in addition kappa >= 1 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-Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb-Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrodinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrodinger equation admits no orthonormal ground state in 1D, for more than one function.

We are interested in the nature of the spectrum of the one-dimensional Schrödinger operator

−d2dx2−Fx+∑n∈Zgnδ(x−n)in L2(R)

with F>0 and two different choices of the coupling constants {gn}n∈Z. In the first model gn≡λ and we prove that if F∈π2Q then the spectrum is R and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model gn are independent random variables with mean zero and variance λ2. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is R and it is dense pure point if F<λ2/2 and purely singular continuous if F>λ2/2.

We consider the fractional Schrodinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces, which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schrodinger operators and have an important application in the treatment of large relativistic atoms.

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=32 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.

Our recent work on the Burchard-Choksi-Topaloglu flocking problem showed that in the large mass regime the ground state density profile is the characteristic function of some set. Here we show that this set is, in fact, a round ball. The essential mathematical structure needed in our proof is a strict rearrangement inequality with a quantitative error estimate, which we deduce from recent deep results of M. Christ.

We discuss a one-dimensional version of the Landau-Pekar equations, which are a system of coupled differential equations with two different time scales. We derive an approximation on the slow time scale in the spirit of a non-linear adiabatic theorem. Dispersive estimates for solutions of the Schrodinger equation with time-dependent potential are a key technical ingredient in our proof. (C) 2020 Elsevier Inc. All rights reserved.

We consider heavy neutral atoms of atomic number Z modeled with kinetic energy (c^2p^2+c^4)^1/2−c^2 used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z,c→∞ keeping Z/c fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.

We consider a large neutral atom of atomic number Z, taking relativistic effects into account by assuming the dispersion relation √(c^2p^2+c^4). We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z,c→∞ keeping Z/c fixed and find that the spherically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the generalization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density.

We show that, within a linear approximation of BCS theory, a weak homogeneous magnetic field lowers the critical temperature by an explicit constant times the field strength, up to higher order terms. This provides a rigorous derivation and generalization of results obtained in the physics literature fromWHH theory of the upper critical magnetic field. A new ingredient in our proof is a rigorous phase approximation to control the effects of the magnetic field.

We study the critical temperature of a superconductive material in a weak external electric potential via a linear approximation of the BCS functional. We reproduce a similar result as in Frank et al. (Commun Math Phys 342(1):189–216, 2016) using the strategy introduced in Frank et al. (The BCS critical temperature in a weak homogeneous magnetic field), where we considered the case of an external constant magnetic field.