Spectral analysis near the low ground energy of magnetic Pauli operators

Diomba Sambou

doi:10.1016/j.crma.2016.02.007

Partial differential equations/Mathematical physics

Spectral analysis near the low ground energy of magnetic Pauli operators

Presented by: the Editorial Board

Diomba Sambou ¹

¹ Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile, Chile

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 606-610.

Abstract
Résumé

We are interested in 3-D magnetic Pauli operators perturbed by a $2 \times 2$ Hermitian matrix-valued potential $V = V (x)$ , $x \in R^{3}$ . We extend to the Pauli case the Breit–Wigner-type approximation and trace formula results obtained for the 3-D Schrödinger operator near the Landau levels. Hence, we give a link between the resonances and the spectral shift function near the low ground energy of the operators.

On s'intéresse à des opérateurs magnétiques 3-D de Pauli perturbés par un potentiel matriciel $2 \times 2$ hermitien $V = V (x)$ , $x \in R^{3}$ . Nous étendons au cas Pauli des résultats d'approximation de type Breit–Wigner et de formule trace obtenus pour l'opérateur de Schrödinger 3-D près des niveaux de Landau. Ainsi, nous établissons un lien entre les résonances et la fonction de décalage spectrale près du bas niveau d'énergie des opérateurs.

Received: 2015-11-08
Accepted: 2016-02-21
Published online: 2016-04-01

DOI: 10.1016/j.crma.2016.02.007

Author's affiliations:

Diomba Sambou ¹

¹ Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile, Chile

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     title = {Spectral analysis near the low ground energy of magnetic {Pauli} operators},
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Diomba Sambou. Spectral analysis near the low ground energy of magnetic Pauli operators. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 606-610. doi : 10.1016/j.crma.2016.02.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.007/

References
Cited by

[1] M.Š. Birman; M.G. Kreĭn On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR, Volume 144 (1962), pp. 475-478 (in Russian); English translation in Sov. Math. Dokl., 3, 1962

[2] J.-F. Bony; V. Bruneau; G. Raikov Resonances and spectral shift function near the Landau levels, Ann. Inst. Fourier, Volume 57 (2007) no. 2, pp. 629-671

[3] J.M. Bouclet Spectral distributions for long range perturbations, J. Funct. Anal., Volume 212 (2004), pp. 431-471

[4] M. Dimassi; J. Sjöstrand Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, UK, 1999

[5] L.S. Koplienko Trace formula for non trace-class perturbations, Sib. Mat. Zh., Volume 35 (1984), pp. 62-71 (in Russian); English transl. in Sib. Math. J., 25, 1984, pp. 735-743

[6] M.G. Krein On the trace formula in perturbation theory, Mat. Sb., Volume 33 (1953), pp. 597-626 (in Russian)

[7] I.M. Lifshits On a problem in perturbation theory, Usp. Mat. Nauk, Volume 7 (1952), pp. 171-180 (in Russian)

[8] G.D. Raikov Low energy asymptotics of the spectral shift function for Pauli operators with nonconstant magnetic fields, Publ. Res. Inst. Math. Sci., Volume 46 (2010), pp. 565-590

[9] D. Sambou Résonances près de seuils d'opérateurs magnétiques de Pauli et de Dirac, Can. J. Math., Volume 65 (2013) no. 5, pp. 1095-1124

[10] J. Sjöstrand A trace formula and review of some estimates for resonances, Microlocal Analysis and Spectral Theory, NATO ASI Series C, vol. 490, Kluwer, 1997, pp. 377-437

[11] J. Sjöstrand Resonances for the bottles and trace formulae, Math. Nachr., Volume 221 (2001), pp. 95-149

[12] D.R. Yafaev Mathematical Scattering Theory. General Theory, Translations of Mathematical Monographs, vol. 105, AMS, Providence, RI, USA, 1992

Cited by Sources:

^☆ This research is partially supported by the Chilean Program Núcleo Milenio de Física Matemática RC120002. The author wishes to express his gratitude to V. Bruneau for suggesting the study of this problem, and thank the anonymous referee for helpful remarks.

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