Rafael Benguria - Academia.edu (original) (raw)

Papers by Rafael Benguria

Partial Differential Equations and Inverse Problems, 2004

To determine the sharp constants for the one dimensional Lieb-Thirring inequalities with exponent... more To determine the sharp constants for the one dimensional Lieb-Thirring inequalities with exponent γ ∈ (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 γ = 1 and an isporimetric inequality for ovals in the plane.

Communications in Mathematical Physics

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in ... more We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of 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. Contents E 5.2. Concavity of the first min-max level 5.3. Proof of the variational principle 6. Geometric upper bounds on the spectral gap 6.1. A simple upper bound 6.2. A sharp upper bound 7. About the Faber-Krahn conjecture 7.1. A new conjecture 7.2. Link with the Bossel-Daners inequality 8. Numerics 8.1. Numerical Methods 8.2. Numerical Results

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Mathematical Research Letters

A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in R d has played a key role in ... more A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in R d has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of R d , in particular for cubes, has arised. The purpose of this manuscript is twofold. 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.

The last unsolved problem about the many-polaron system, in the Pekar-Tomasevich approximation, i... more 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 .

Transactions of the American Mathematical Society, 2000

For a domain Ω \Omega contained in a hemisphere of the n n –dimensional sphere S n \mathbb {S}^n ... more For a domain Ω \Omega contained in a hemisphere of the n n –dimensional sphere S n \mathbb {S}^n we prove the optimal result λ 2 / λ 1 ( Ω ) ≤ λ 2 / λ 1 ( Ω ⋆ ) \lambda _2/\lambda _1(\Omega ) \le \lambda _2/\lambda _1(\Omega ^{\star }) for the ratio of its first two Dirichlet eigenvalues where Ω ⋆ \Omega ^{\star } , the symmetric rearrangement of Ω \Omega in S n \mathbb {S}^n , is a geodesic ball in S n \mathbb {S}^n having the same n n –volume as Ω \Omega . We also show that λ 2 / λ 1 \lambda _2/\lambda _1 for geodesic balls of geodesic radius θ 1 \theta _1 less than or equal to π / 2 \pi /2 is an increasing function of θ 1 \theta _1 which runs between the value ( j n / 2 , 1 / j n / 2 − 1 , 1 ) 2 (j_{n/2,1}/j_{n/2-1,1})^2 for θ 1 = 0 \theta _1=0 (this is the Euclidean value) and 2 ( n + 1 ) / n 2(n+1)/n for θ 1 = π / 2 \theta _1=\pi /2 . Here j ν , k j_{\nu ,k} denotes the k k th positive zero of the Bessel function J ν ( t ) J_{\nu }(t) . This result generalizes the Payne–Pólya–W...

After completing my Ph.D. in Physics at Princeton University, I obtained a postdoctoral position ... more After completing my Ph.D. in Physics at Princeton University, I obtained a postdoctoral position at Rockefeller University. In the early sixties, Detlev Bronk, who was president (1953–1968) of the then Rockefeller Institute for Medical Research hired George Uhlenbeck, Mark Kac and Theodore Berlin (1961), among others, to establish a Mathematical Physics group, and Abraham Pais (1962) to lead a group in High Energy Physics. In fact Detlev Bronk successfully made the transition from a research institute to the Rockefeller University (1965). Kenneth Case joined the Mathematical Physics group at Rockefeller in 1969, and so did James Glimm in 1974. I stayed in Kenneth Case’s Lab at Rockefeller University from 1979 to 1981. There I had the chance to work with Ken Case and Mark Kac [2], to meet many visitors in Mathematics and Physics and to enjoy the friendly atmosphere of the 14th floor of the Tower Building where the labs of Ken Case and Eddie Cohen were housed. This year marks the hund...

Bulletin of the London Mathematical Society, 2021

In this paper we analyse the spectrum of nonlocal Dirichlet problems with nonsingular kernels in ... more In this paper we analyse the spectrum of nonlocal Dirichlet problems with nonsingular 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.

Nonlinear Analysis, 2017

We consider the Brezis-Nirenberg problem for the Laplacian with a singular drift for a (geodesic)... more We consider the Brezis-Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both R n and S n , 3 ≤ n ≤ 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.

Journal of Mathematical Physics, 2015

The last unsolved problem about the many-polaron system, in the Pekar-Tomasevich approximation, i... more 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 .

Physical Review E, 2007

We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction... more We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction-diffusion equation when a small cutoff is applied to the reaction term at the unstable or metastable equilibrium point. The results are valid for arbitrary reaction terms and include the case of density-dependent diffusion.

Physical Review E, 2004

We study the minimal speed of propagating fronts of convection-reaction-diffusion equations of th... more We study the minimal speed of propagating fronts of convection-reaction-diffusion equations of the form u t ϩ(u)u x ϭu xx ϩ f (u) for positive reaction terms with f Ј(0)Ͼ0. The function (u) is continuous and vanishes at uϭ0. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assessment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if f Љ(u)/ͱf Ј(0)ϩЈ(u)Ͻ0, then the minimal speed is given by the linear value 2ͱf Ј(0), and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case u t ϩuu x ϭu xx ϩu(1Ϫu). Results are also given for the density dependent diffusion case u t ϩ(u)u x ϭ͓D(u)u x ͔ x ϩ f (u).

Oberwolfach Reports, 2009

Journal of Physics A: Mathematical and Theoretical, 2012

For the case of the bipolaron, it has been recently proved that for U ≥ 53.2α, where U is the rep... more For the case of the bipolaron, it has been recently proved that for U ≥ 53.2α, where U is the repulsion parameter of the electrons, and α is the coupling constant of the polaron, no binding occurs. We show that actually for U ≥ 52.1α, there is no binding. Furthermore, we obtain special results for small and large values of α: more specifically, we prove that for each ε > 0, there is an α 1 and an α 2 , such that if 0 < α ≤ α 1 , a condition for no-binding becomes U ≥ (40.4 + ε)α, and if α ≥ α 2 , it is U ≥ (38.7 + ε)α. Here, α 1 can be computed with any desired accuracy, whereas we were only able to prove the existence of such an α 2 .

Journal of Mathematical Physics, 2011

An explicit asymptotic expression for the ground-state energy of the Pekar-Tomasevich functional ... more An explicit asymptotic expression for the ground-state energy of the Pekar-Tomasevich functional for the N-polaron is found, when the repulsion parameter U of the electrons satisfies the inequality 0 ≤ U ≤ 2α, where α is the coupling constant of the polaron. If E N U denotes this ground-state energy for the case of N electrons and repulsion parameter U , we prove that E N U /N 3 → −c p (U − 2α) 2 /4 as N → ∞, where, c p = 0.10851. .. . Moreover, we show that E N 0 = −c p α 2 N 3 , for all N .

Journal of Mathematical Physics, 2012

In this manuscript we provide a family of lower bounds on the indirect Coulomb energy for atomic ... more 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.

Journal of Functional Analysis, 2009

We consider a quantity κ(Ω)-the distance to the origin from the null variety of the Fourier trans... more We consider a quantity κ(Ω)-the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximized, among all convex balanced domains Ω ⊂ R d of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians.

Journal of Differential Equations, 2004

International Journal of Quantum Chemistry, 2011

Here we prove a new lower bound on the indirect Coulomb energy in quantum mechanics, in terms of ... more Here 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, C = 1.45 < C LO = 1.68) but involving an additive kinetic energy term of the single particle density as well.

Duke Mathematical Journal, 2007

Let Ω be some domain in the hyperbolic space H n (with n ≥ 2) and S 1 the geodesic ball that has ... more Let Ω be some domain in the hyperbolic space H n (with n ≥ 2) and S 1 the geodesic ball that has the same first Dirichlet eigenvalue as Ω. We prove the Payne-Pólya-Weinberger conjecture for H n , 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.

Communications in Mathematical Physics, 2006

Let λ i (Ω, V) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditio... more Let λ i (Ω, V) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω ⊂ 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 ⋆ , we prove that λ 2 (Ω, V) ≤ λ 2 (S 1 , V ⋆). Here S 1 denotes the ball, centered at the origin, that satisfies the condition λ 1 (Ω, V) = λ 1 (S 1 , V ⋆). 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.