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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-03702964
Contributor : Maria J. Esteban Connect in order to contact the contributor
Submitted on : Thursday, June 23, 2022 - 3:02:20 PM
Last modification on : Wednesday, June 29, 2022 - 3:46:49 AM

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  • HAL Id : hal-03702964, version 1
  • ARXIV : 2206.11679

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Jean Dolbeault, Maria J Esteban, Eric Séré. Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators. 2022. ⟨hal-03702964⟩

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