Essential Self-Adjointness and Invariance of the Essential Spectrum for Dirac Operators

Arai, Masato; Yamada, Osanobu

doi:10.2977/prims/1195183289

Cited by 21 publications

(17 citation statements)

References 10 publications

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“…. As for the proof of (5.3), by means of the short distance asymptotics (4.2) for v 0 and v ∞ we find The next preparatory step is to introduce, for later convenience, the Wronskian of any two square-integrable functions, 4) and the boundary form for any two functions in D(S * ),…”

Section: )mentioning

confidence: 97%

Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei

Gallone

Michelangeli

2018

Anal.Math.Phys.

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We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kreȋn-Višik-Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.Mathematics Subject Classification (2010). 47B25; 47N20; 47N50; 81Q10.2 , n ∈ N 0 , κ ∈ Z\{0} .(1.12)Although the above regime of Z covers all currently known elements (the last one to be discovered, the Oganesson 294 118 Og, thus Z = 118, was first synthesized in 2002 and formally named in 2016), the problem of the self-adjoint realisation of the Dirac-Coulomb Hamiltonian above the threshold Zα f = √ 3 2 has been topical since long and so is still today. Even the consideration that the problem only arises due to the idealisation of point-like nuclei (and also because one neglects the anomalous magnetic moment of the electron) does not diminish its relevance, given the extreme experimental precision, for example, of Sommerfeld's fine-structure formula for the eigenvalues of H when Z 118.From the mathematical side, the study of the self-adjoint extensions of the Dirac-Coulomb Hamiltonian has a long and active history

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Section: )mentioning

confidence: 97%

Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei

Gallone

Michelangeli

2018

Anal.Math.Phys.

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“…The problem of essential selfadjointness of the Dirac operator, which is a special case of our differential operator, has many references; see, for example, [1, 4, 14, 16, 26, 28, 32] and papers quoted therein. Usually (see, [1,14,16,32]) functions bounded at infinity but with poles in R m are considered. The proof of selfadjointness of A relies in such cases on showing that A 1 u d A 0 u + d u (u ∈ (C ∞ 0 (R m )) k ) for some d ∈ [0, 1), d 0 (for example, [16, Problem V.5.11]) or d ∈ [0, 1), d = 0 (for example, [14]).…”

Section: We Letmentioning

confidence: 99%

A criterion for selfadjointness in Krein spaces

Wojtylak

2008

Bulletin of the London Mathematical Society

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“…The papers [15,14,16,17,18,13,12,10] treated this problem by using a different method depending on the singularity of the potential. Those works dealt exclusively with electrostatic potentials, while in [2,3,4,20], Arai and Yamada consider more general matrix-valued potentials. Results on the essential self-adjointness of Dirac operators with relativistic δ-sphere interactions can be found in [5,6,8] and similar results for the Schrödinger operator with point interactions in [1].…”

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confidence: 99%