Spectral Analysis of Relativistic Operators

Balinsky, Alexander; Evans, W. D.

doi:10.1142/p566

Cited by 30 publications

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“…on the Hilbert space Λ + L 2 (R 3 ; C 4 ). For a comprehensive review we refer to the textbook of Balinsky and Evans [2]. While the physically relevant case is γ > 0, we are interested in the case where γ = −ν ∈ [−1, 0].…”

Section: The Self-adjoint Extension Constructed In This Way Coincidesmentioning

confidence: 99%

Friedrichs Extension and Min–Max Principle for Operators with a Gap

Schimmer

Solovej

Tokus

2019

Ann. Henri Poincaré

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Semibounded symmetric operators have a distinguished self-adjoint extension, the Friedrichs extension. The eigenvalues of the Friedrichs extension are given by a variational principle that involves only the domain of the symmetric operator. Although Dirac operators describing relativistic particles are not semibounded, the Dirac operator with Coulomb potential is known to have a distinguished extension. Similarly, for Dirac-type operators on manifolds with a boundary a distinguished self-adjoint extension is characterised by the Atiyah-Patodi-Singer boundary condition. In this paper we relate these extensions to a generalisation of the Friedrichs extension to the setting of operators satisfying a gap condition. In addition we prove, in the general setting, that the eigenvalues of this extension are also given by a variational principle that involves only the domain of the symmetric operator.for k ≥ 1 are the discrete spectrum of A F in the interval (−∞, sup k≥1 λ k ), counted with multiplicities d k := # {j ≥ 1 : λ j = λ k } 2010 Mathematics Subject Classification. 49R05, 49S05, 47B25, 81Q10.

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Section: The Self-adjoint Extension Constructed In This Way Coincidesmentioning

confidence: 99%

Friedrichs Extension and Min–Max Principle for Operators with a Gap

Schimmer

Solovej

Tokus

2019

Ann. Henri Poincaré

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“…Using the scalar version of Lemma 2.10 again and that H α0 0 is the Friedrichs extension of H α0 0 we conclude (3.5). The following description of the domains of H α l with α α l follows analogously to Theorem 2 and Corollary 2 in [11] (see also Section 2.2.3 of [4]), where the case l = 0 is studied.…”

Section: )mentioning

confidence: 99%

Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications

Morozov

Muller

2017

Journal of Mathematical Physics

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For ν ∈ [0, 1] let D ν be the distinguished self-adjoint realisation of the three-dimensional Coulomb-Dirac operator −iα · ∇ − ν| · | −1 . For ν ∈ [0, 1) we prove the lower bound of the form |D ν | Cν √ −∆, where Cν is found explicitly and is better then in all previous works on the topic. In the critical case ν = 1 we prove that for every λ ∈ [0, 1) there exists K λ > 0 such that the estimate |D 1 | K λ a λ−1 (−∆) λ/2 − a −1 holds for all a > 0. As applications we extend the range of coupling constants in the proof of the stability of the relativistic electron-positron field and obtain Cwickel-Lieb-Rozenblum and Lieb-Thirring type estimates on the negative eigenvalues of perturbed projected massless Coulomb-Dirac operators in the Furry picture. We also study the existence of a virtual level at zero for such projected operators.

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“…The estimate is optimal in the sense that (3) implies that V ∈ L d/2 (R d ) (see, e.g., [37, (127)]). It is well known that (2) does not hold for d = 2.…”

Section: Introductionmentioning

confidence: 99%

On estimates for the number of negative eigenvalues of two-dimensional Schrödinger operators with potentials supported by Lipschitz curves

Karuhanga

2017

Journal of Mathematical Analysis and Applications

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Spectral Analysis of Relativistic Operators

Cited by 30 publications

References 0 publications

Friedrichs Extension and Min–Max Principle for Operators with a Gap

Friedrichs Extension and Min–Max Principle for Operators with a Gap

Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications

On estimates for the number of negative eigenvalues of two-dimensional Schrödinger operators with potentials supported by Lipschitz curves

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