Resonances and Spectral Shift Function near the Landau levels
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, p. 629-671
We consider the 3D Schrödinger operator H=H 0 +V where H 0 =(-i-A) 2 -b, A is a magnetic potential generating a constant magneticfield of strength b>0, and V is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of H admits a meromorphic extension from the upper half plane to an appropriate Riemann surface , and define the resonances of H as the poles of this meromorphic extension. We study their distribution near any fixed Landau level 2bq, q. First, we obtain a sharp upper bound of the number of resonances in a vicinity of 2bq. Moreover, under appropriate hypotheses, we establish corresponding lower bounds which imply the existence of an infinite number of resonances, or the absence of resonances in certain sectors adjoining 2bq. Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair (H,H 0 ) as a sum of a harmonic measure related to the resonances, and the imaginary part of a holomorphic function. This representation justifies the Breit-Wigner approximation, implies a trace formula, and provides information on the singularities of the SSF at the Landau levels.
On étudie l’opérateur de Schrödinger magnétique en dimension 3, H=H 0 +VH 0 =(-i-A) 2 -b, A est un potentiel magnétique générant un champ magnétique constant de force b>0 fixée et V est un potentiel électrique qui décroît super-exponentiellement dans la direction du champ magnétique. On montre que la résolvante de H admet un prolongement méromorphe du plan supérieur une certaine surface de Riemann et on définit les résonances de H comme les pôles de cette extension méromorphe. On étudie leur répartition près d’un niveau de Landau fixé 2bq, q. On obtient d’abord des majorations du nombre de résonances dans des petits domaines proches de 2bq. Sous des hypothses supplémentaires, on prouve des minorations du nombre de résonances qui implique la présence d’une infinité de résonances ou bien l’absence de résonances dans certains secteurs de sommet 2bq. Finalement, on montre que la fonction de décalage spectral (FDS) associée à la paire (H,H 0 ) est la somme de mesures harmoniques associées aux résonances et de la partie imaginaire d’une fonction holomorphe. Cette formule justifie l’approximation de Breit-Wigner, implique une formule de trace à la Sjöstrand et donne des informations sur les singularités de la FDS aux niveaux de Landau.
DOI : https://doi.org/10.5802/aif.2270
Classification:  35P25,  35J10,  47F05,  81Q10
Keywords: Magnetic Schrödinger operators, resonances, spectral shift function, Breit-Wigner approximation
@article{AIF_2007__57_2_629_0,
     author = {Bony, Jean-Fran\c cois and Bruneau, Vincent and Raikov, Georgi},
     title = {Resonances and Spectral Shift Function near the Landau levels},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     pages = {629-671},
     doi = {10.5802/aif.2270},
     zbl = {1129.35053},
     mrnumber = {2310953},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_2_629_0}
}
Bony, Jean-François; Bruneau, Vincent; Raikov, Georgi. Resonances and Spectral Shift Function near the Landau levels. Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 629-671. doi : 10.5802/aif.2270. https://aif.centre-mersenne.org/item/AIF_2007__57_2_629_0/

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