Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math-ph/0309065

An analytical proof of Hardy-like inequalities related to the Dirac operator

J. Dolbeault·M. J. Esteban·M. Loss·L. Vega
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.

Explore related subjects

math-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

J. Dolbeault, M. J. Esteban, M. Loss, L. Vega. 2003-09-30. An analytical proof of Hardy-like inequalities related to the Dirac operator. https://arxiv.org/abs/math-ph/0309065

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Massless Representations in Conformal Space and Their de Sitter Restrictions

The monograph offers a coherent and self-contained treatment of massless (ladder) representations of the conformal group U(2,2) and their restriction to the de Sitter group Sp(2,2), combining rigorous representation-theoretic analysis with fully explicit constructions. It systematically develops these representations, including the derivation of invariant bilinear forms and Casimir operators, and constructs vertex operators and two-point functions for low-helicity fields. A central and distinctive contribution is the introduction of a canonical Clifford-split-octonion framework, in which 8-component Majorana spinors are realized within an alternative composition algebra, providing a unified and intrinsically defined setting for the algebraic, spinorial, and geometric structures underlying the theory. By bridging abstract symmetry principles with concrete computational methods and physically motivated applications in quantum field theory and cosmology, the monograph advances both conceptual clarity and technical control. While primarily addressed to researchers in mathematical physics and related fields, the exposition is carefully structured to guide advanced graduate students through subtle constructions, maintaining accessibility without compromising mathematical precision.

math-ph↗
arXiv2026

Gauge invariant generalizations of the Proca equation and the Yang-Mills-Proca equation

The Proca equations (1936) are used in quantum field theory to describe vector bosons (spin 1) with a nonzero mass. The Proca equations are not gauge invariant. In contrast to Stueckelberg's approach (1938), this article presents a gauge invariant generalization of the Proca equation by introducing an additional vector field into the Proca equation. The results are extended to the Yang-Mills-Proca equations, leading to equations with non-Abelian gauge symmetry.

math-ph↗
arXiv2026

A Wigner-Lindblad phase space model for open quantum dynamics

We investigate the collisional quantum dynamics of an open system in contact with a thermal bath. We propose a particle-bath interaction model based on the Weyl quantization principle and the Wigner phase space description of quantum evolution. By factorizing the reduced Hamiltonian of the open system via supersymmetric quantum mechanics techniques, we construct system-dependent creation and annihilation operators for arbitrary potential landscapes. This allows us to extend the classical Fokker--Planck relaxation mechanism to incorporate Lindblad quantum stochastic jumps within the Wigner phase-space formalism. Through a semiclassical expansion of the Moyal star product to second order, we obtain a generalized Wigner--Lindblad kinetic model featuring position-dependent friction and diffusion terms. We establish the mathematical well-posedness of the Lindblad problem and we validate our approach through numerical simulations. Our model resolves unphysical long-time artifacts of standard quantum Fokker--Planck models in quartic double-well traps, accurately captures dissipative quantum state localization in 2D molecular potentials (4D phase space), and effectively evaluates electron transport and trapping times in InAs/InP resonant tunneling diodes.

math-ph↗