arXiv subjects
Publications and source records attributed to Rafael D. Benguria.
We prove an upper bound on the excess charge for non-relativistic atomic systems, independent of the particle statistics. This result improves Lieb's bound of 1984 for any $Z \ge 12$.
In this manuscript, using a technique introduced by P.~T.~Nam in 2012 and the {\it Coulomb Uncertainty Principle}, we prove new bounds on the excess charge for non relativistic atomic systems, independent of the particle statistics. These new bounds are the best bounds to date for bosonic systems for all values of the atomic number $Z$ and they are also the best bounds for fermionic systems with $Z \le 26$ (i.e., up to the chemical element {\it iron})
We bound the number of electrons $Q$ that an atom can bind in excess of neutrality for density functionals generalizing the classical Thomas-Fermi-Weizsäcker functional: instead of the classical power $5/3$ more general powers $p$ are considered. For $3/2<p<2$ we prove the excess charge conjecture, i.e., that $Q$ is uniformly bounded in the atomic number $Z$. The case $p=3/2$ is critical: the behavior changes from a uniform bound in $Z$ to a linear bound at the critical coupling $4\sqrtπ$ of the nonlinear term. We also improve the linear bound for all $p\geq6/5$.
In this paper we obtain a Hadamard type formula for simple eigenvalues and an analog to the Rayleigh-Faber-Krahn inequality for a class of nonlocal eigenvalue problems. Such class of equations include among others, the classical nonlocal problems with Dirichlet and Neumann conditions. The Hadamard formula is computed allowing domain perturbations given by embeddings of $n$-dimensional Riemannian manifolds (possibly with boundary) of finite volume while the Rayleigh-Faber-Krahn inequality is shown by rearrangement techniques.
We consider four-component Dirac operators on domains in the plane. With suitable boundary conditions, these operators describe graphene quantum dots. The most general boundary conditions are defined by a matrix depending on four real parameters. For operators with constant boundary parameters we show that the Hamiltonian is unitary equivalent to two copies of the two-component operator. This allows to extend the known results for this type of operators to the four-component case. As an application, we identify the boundary conditions from the tight-binding model for graphene that give rise to a block-diagonal operator in the continuum limit.
We prove an analytic bound on the excess charge for the Hartree equation in the atomic case.
We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegö type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.
In this paper we analyse the spectrum of nonlocal Dirichlet problems with non-singular kernels in bounded open sets. The novelty is the continuity of eigenvalues with respect to domain perturbation via Lebesgue measure. Also, under additional smooth condition on the kernel and domain, we prove differentiability of simple eigenvalues computing their first derivative.
Inspired by the seminal, ground-breaking work of Abrikosov in 1957, we developed a new approximation to the interaction between two widely separated superconducting vortices. In contrast with Abrikosov's, we take into account the finite size of the vortices and their internal magnetic profile. We consider the vortices to be embedded within a superconducting, infinitely long hollow cylinder, in order to simplify the symmetry and boundary conditions for the mathematical analysis. We study this system in the context of a magnetic Ginzburg-Landau functional theory, by solving for the magnetic field profile inside each vortex, as well as in the superconducting region, subject to physical boundary conditions inspired by the classical analogue of two mutually inducting coils. Under isothermal conditions, the effective force between these vortices is given by the gradient of the Helmholtz free energy constructed from the Ginzburg-Landau functional. From our results, we explicitly show that, in agreement with well established theoretical arguments and experiments, the interaction between widely separated vortices is repulsive in this context, and their equilibrium positions are constrained by the fluxoid's conservation. Moreover, we find that the equilibrium positions of the vortices centers are stable due to the convexity of the Helmholtz free energy profile. Remarkably, the effect of the boundaries of the region over the effective interaction between the vortices is important in the chosen geometric configuration.
We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le n \le 5$. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ($δ$ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space is characterized by the (geodesic) radius of the ball, $δ$, and $λ$, the coupling constant of the linear term of the Brezis-Nirenberg problem.
We develop a new method for estimating the region of the spectral parameter of a generalized Brezis--Nirenberg problem for which there are no, non trivial, smooth solutions. This new method combines the standard Rellich--Pohozaev argument with a Hardy type inequality for bounded domains. The estimates we get are better than the usual estimates for low dimensions.
In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincaré-Sobolev inequalities. We show that minimizers do exist for smooth domains in $\mathbb{R}^d$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in $\mathbb{R}^2$.
Here we prove an isoperimetric inequality for the harmonic mean of the first $N-1$ non-trivial Neumann eigenvalues of the Laplace-Beltrami operator for domains contained in a hemisphere of $\mathbb{S}^N$.
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.
We consider Dirac operators defined on planar domains. For a large class of boundary conditions, we give a direct proof of their self-adjointness in the Sobolev space $H^1$.
The two-dimensional Dirac operator describes low-energy excitations in graphene. Different choices for the boundary conditions give rise to qualitative differences in the spectrum of the resulting operator. For a family of boundary conditions, we find a lower bound to the spectral gap around zero, proportional to $|Ω|^{-1/2}$, where $Ω\subset \mathbb{R}^2$ is the bounded region where the Dirac operator acts. This family contains the so-called infinite mass and armchair cases used in the physics literature for the description of graphene quantum dots.
We consider the Pauli operator in $\mathbb R^3$ for magnetic fields in $L^{3/2}$ that decay at infinity as $|x|^{-2-β}$ with $β> 0$. In this case we are able to prove that the existence of a zero mode for this operator is equivalent to a quantity $δ(\mathbf B)$, defined below, being equal to zero. Complementing a result from [Balinsky, Evans, Lewis (2001)], this implies that for the class of magnetic fields considered, Sobolev, Hardy and CLR inequalities hold whenever the magnetic field has no zero mode.
The last unsolved problem about the many-polaron system, in the Pekar-Tomasevich approximation, is the case of bosons with the electron-electron Coulomb repulsion of strength exactly 1 (the 'neutral case'). We prove that the ground state energy, for large $N$, goes exactly as $-N^{7/5}$, and we give upper and lower bounds on the asymptotic coefficient that agree to within a factor of $2^{2/5}$.