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Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators

Dolbeault, Jean; Esteban, Maria J.; Séré, Eric (2022), Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators. https://basepub.dauphine.psl.eu/handle/123456789/23102

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DES-2022.pdf (454.2Kb)
Type
Document de travail / Working paper
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
17
Metadata
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Author(s)
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Esteban, Maria J. cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Séré, Eric
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the minimal Dirac-Coulomb operator defined on C ∞ c (R 3 \{0}, C 4). In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a variational min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that by making only minimal assumptions. Our result applied to the Dirac-Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137.
Subjects / Keywords
variational methods; self-adjoint operators; quadratic forms; spectral gaps; eigenvalues; min-max principle; Rayleigh-Ritz quotients; Dirac operators

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