arXiv subjects

Albert Mas

Publications and source records attributed to Albert Mas.

26 records · Page 2Linked to original sources

arXiv2015

Dirac operators, shell interactions and discontinuous gauge functions across the boundary

Given a bounded smooth domain $Ω\subset\mathbb{R}^3$, we explore the relation between couplings of the free Dirac operator $-iα\cdot\nabla+mβ$ with pure electrostatic shell potentials $λδ_{\partialΩ}$ ($λ\in\mathbb{R}$) and some perturbations of those potentials given by the normal vector field $N$ on the shell $\partialΩ$, namely $\{λ_e+λ_n(α\cdot N)\}δ_{\partialΩ}$ ($λ_e$, $λ_n\in\mathbb{R}$). Under the appropiate change of parameters, the couplings with perturbed and unperturbed electrostatic shell potentials yield unitary equivalent self-adjoint operators. The proof relies on the construction of an explicit family of unitary operators that is well adapted to the study of shell interactions, and fits within the framework of gauge theory. A generalization of such unitary operators also allow us to deal with the self-adjointness of couplings of $-iα\cdot\nabla+mβ$ with some shell potentials of magnetic type, namely $λ(α\cdot N)δ_{\partialΩ}$ with $λ\in C^1(\partialΩ)$.

math-ph↗

arXiv2015

An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators

In this article we investigate spectral properties of the coupling $H+V_λ$, where $H=-iα\cdot\nabla +mβ$ is the free Dirac operator in $\mathbb R^3$, $m>0$ and $V_λ$ is an electrostatic shell potential (which depends on a parameter $λ\in\mathbb R$) located on the boundary of a smooth domain in $\mathbb R^3$. Our main result is an isoperimetric-type inequality for the admissible range of $λ$'s for which the coupling $H+V_λ$ generates pure point spectrum in $(-m,m)$. That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman-Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible $λ$'s, and we use this to relate the endpoints of the admissible range of $λ$'s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.

math-ph↗

arXiv2014

Shell interactions for Dirac operators: on the point spectrum and the confinement

Spectral properties and the confinement phenomenon for the coupling $H+V$ are studied, where $H=-iα\cdot\nabla +mβ$ is the free Dirac operator in $\mathcal{R}^3$ and $V$ is a measure-valued potential. The potentials $V$ under consideration are given in terms of surface measures on the boundary of bounded regular domains in $\mathcal{R}^3$. A criterion for the existence of point spectrum is given, with applications to electrostatic shell potentials. In the case of the sphere, an uncertainty principle is developed and its relation with some eigenvectors of the coupling is shown. Furthermore, a criterion for generating confinement is given. As an application, some known results about confinement on the sphere for electrostatic plus Lorentz scalar shell potentials are generalized to regular surfaces.

math.AP↗

arXiv2013

Shell interactions for Dirac operators

The self-adjointness of $H+V$ is studied, where $H=-iα\cdot\nabla +mβ$ is the free Dirac operator in $\R^3$ and $V$ is a measure-valued potential. The potentials $V$ under consideration are given by singular measures with respect to the Lebesgue measure, with special attention to surface measures of bounded regular domains. The existence of non-trivial eigenfunctions with zero eigenvalue naturally appears in our approach, which is based on well known estimates for the trace operator defined on classical Sobolev spaces and some algebraic identities of the Cauchy operator associated to $H$.

math.AP↗

arXiv2011

Variation for singular integrals on Lipschitz graphs: L^p and endpoint estimates

Let 0 2, we prove that the r-variation and oscillation for Calderón-Zygmund singular integrals with odd kernel are bounded operators in L^p(H) for 1<p finite, from L^1(H) to weak-L^1(H), and from the space of bounded H-measurable functions to BMO(H). Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in R^d to weak-L^1(H).

math.CA↗

arXiv2011

Variation for the Riesz transform and uniform rectifiability

For 0 2, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded operator in L^2(M). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L^2(M) boundedness of the Riesz transform to the uniform rectifiability of M.

math.CA↗

arXiv2011

Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs

We prove that, for r>2, the r-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in L^p for 1<p finite. The analogous result holds for the n-dimensional Riesz transform on n-dimensional Lipschitz graphs, as well as for other singular integral operators with odd kernel. In particular, our results strengthen the classical theorem on the L^2 boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.

math.CA↗

arXiv2010

A dual characterization of the C^1 harmonic capacity and applications

The Lipschitz and C^1 harmonic capacities K and K_c in R^n can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities G and A (respectively). In this paper we provide a dual characterization of K_c in the spirit of the classical one for the capacity A by means of the Garabedian function.

math.AP↗