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Shell Interactions for Dirac Operators: On the Point Spectrum and the Confinement

Shell interactions for Dirac operators: on the point spectrum and the confinement
Authors: Arrizabalaga, Naiara; Mas Blesa, Albert; Vega, Luis;

Shell Interactions for Dirac Operators: On the Point Spectrum and the Confinement

Abstract

Spectral properties and the confinement phenomenon for the coupling $H+V$ are studied, where $H=-i��\cdot\nabla +m��$ 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.

25 pages, 2 figures

Keywords

singular integral, Covariant wave equations in quantum theory, relativistic quantum mechanics, self-adjoint extension, Dirac operator, Mathematical analysis, Atomic physics, shell interaction, Mathematics - Analysis of PDEs, Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica, Shell interaction, Anàlisi matemàtica, :Matemàtiques i estadística::Anàlisi matemàtica [Àrees temàtiques de la UPC], FOS: Mathematics, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Self-adjoint extension, Singular integral, Selfadjoint operator theory in quantum theory, including spectral analysis, PDEs in connection with quantum mechanics, Analysis of PDEs (math.AP)

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selected citations
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This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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