ASTABURUAGA, Maria Angélica; BRIET, Philippe; BRUNEAU, Vincent; FERNANDEZ, Claudio; RAIKOV, Georgi

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ASTABURUAGA, Maria Angélica
Departamento de Matemáticas [Santiago de Chile]

BRIET, Philippe
Centre de Physique Théorique - UMR 6207 [CPT]
Centre de Physique Théorique - UMR 7332 [CPT]

BRUNEAU, Vincent
Institut de Mathématiques de Bordeaux [IMB]

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Ce document a été publié dans

Serdica Mathematical Journal. 2008, vol. 34, p. 179-218

Bulgarian Academy of Sciences, Institute of Mathematics

Résumé en anglais

We consider the Hamiltonian $H$ of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator $H$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H$ by appropriate scalar potentials $V$ and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant $\varkappa$ of the perturbation tends to zero. Further, under the assumption that the Fermi Golden Rule holds true, we deduce estimates for the time evolution of the resonance states with and without analyticity assumptions; in the second case we obtain these results as a corollary of suitable Mourre estimates and a recent article of Cattaneo, Graf and Hunziker \cite{cgh}. Next, we describe sets of perturbations $V$ for which the Fermi Golden Rule is valid at each embedded eigenvalue of $H$; these sets turn out to be dense in various suitable topologies. Finally, we assume that $V$ decays fast enough at infinity and is of definite sign, introduce the Krein spectral shift function for the operator pair $(H+V, H)$, and study its singularities at the energies which coincide with eigenvalues of infinite multiplicity of the unperturbed operator $H$.< Réduire

Mots clés en anglais

magnetic Schroedinger operators

resonances

Mourre estimates

spectral shift function

URI

https://oskar-bordeaux.fr/handle/20.500.12278/191571

Project ANR

Equations hyperboliques dans des espaces-temps de la relativité générale : diffusion et résonances. - ANR-05-JCJC-0087

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Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251