arXiv subjects
Publications and source records attributed to Rafael D. Benguria.
In this manuscript we provide a family of lower bounds on the indirect Coulomb energy for atomic and molecular systems in two dimensions in terms of a functional of the single particle density with gradient correction terms.
We prove a new lower bound on the indirect Coulomb energy in two dimensional quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the Lieb--Solovej--Yngvason bound with a smaller constant, C=(4/3)^{3/2}\sqrt{5 π-1} \approx 5.90 < C_{LSY}= 192\sqrt{2 π} \approx 481.27, which also involves an additive gradient energy term of the single particle density.
We prove a new lower bound on the indirect Coulomb energy in quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the classical Lieb--Oxford bound (with a smaller constant, 1.45 < 1.68) but involving an additive kinetic energy term of the single particle density.
We study the effect of a cut-off on the speed of pulled fronts of the one dimensional reaction diffusion equation. We prove rigorous upper and lower bounds on the speed in terms of the cut-off parameter epsilon. From these bounds we estimate the range of validity of the Brunet--Derrida formula for a general class of reaction terms.
We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsaecker model of atoms and molecules. We find bounds for the critical nuclear charges that ensure stability.
It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the three dimensional upper half space is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well.
Let $Ω$ be some domain in the hyperbolic space $\Hn$ (with $n\ge 2$) and $S_1$ the geodesic ball that has the same first Dirichlet eigenvalue as $Ω$. We prove the Payne-Pólya-Weinberger conjecture for $\Hn$, i.e., that the second Dirichlet eigenvalue on $Ω$ is smaller or equal than the second Dirichlet eigenvalue on $S_1$. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius.
Let $λ_i(Ω,V)$ be the $i$th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $Ω\subset \R^n$ and with the positive potential $V$. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\star$, we prove that $λ_2(Ω,V) \le λ_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $λ_1(Ω,V) = λ_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $λ_2(B_R, V) / λ_1(B_R, V)$ decreases when the radius $R$ of the ball $B_R$ increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.
For the Pauli-Fierz operator with a short range potential we study the binding threshold as a function of the fine structure constant $α$ and show that it converges to the binding threshold for the Schrödinger operator in the small $α$ limit.
To determine the sharp constants for the one dimensional Lieb--Thirring inequalities with exponent gamma in (1/2,3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be attained by potentials having only one bound state. Here we exhibit a connection between the Lieb--Thirring conjecture for gamma=1 and an isporimetric inequality for ovals in the plane.
For a domain $Ω$ contained in a hemisphere of the $n$-dimensional sphere $\SS^n$ we prove the optimal result $λ_2/λ_1(Ω) \le λ_2/λ_1(Ω^{\star})$ for the ratio of its first two Dirichlet eigenvalues where $Ω^{\star}$, the symmetric rearrangement of $Ω$ in $\SS^n$, is a geodesic ball in $\SS^n$ having the same $n$-volume as $Ω$. We also show that $λ_2/λ_1$ for geodesic balls of geodesic radius $θ_1$ less than or equal to $π/2$ is an increasing function of $θ_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ for $θ_1=0$ (this is the Euclidean value) and $2(n+1)/n$ for $θ_1=π/2$. Here $j_{ν,k}$ denotes the $k^{th}$ positive zero of the Bessel function $J_ν(t)$. This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of $\SS^n$ and having a fixed value of $λ_1$ the one with the maximal value of $λ_2$ is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for $λ_2/λ_1$. Various other results for $λ_1$ and $λ_2$ of geodesic balls in $\SS^n$ are proved in the course of our work.