Resonances and Spectral Shift Function near the Landau levels

Bony, Jean-François; Bruneau, Vincent; Raikov, Georgi

doi:10.5802/aif.2270

Resonances and Spectral Shift Function near the Landau levels
[Résonances et fonction de décalage spectral près des niveaux de Landau]

Bony, Jean-François ¹ ; Bruneau, Vincent ¹ ; Raikov, Georgi ²

¹ Université Bordeaux I FR CNRS 2254, MAB UMR CNRS 5466 Institut de Mathématiques de Bordeaux 351 cours de la Libération 33405 Talence (France)
² Pontificia Universidad Católica de Chile Facultad de Matemáticas Departamento de Matemáticas Vicuña Mackenna 4860 Santiago de Chile (Chile)

Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 629-671.

Résumé
Abstract

On étudie l’opérateur de Schrödinger magnétique en dimension 3, $H = H_{0} + V$ où $H_{0} = {(- i \nabla - A)}^{2} - b$ , $A$ 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 $ℳ$ 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 $2 b q$ , $q \in ℕ$ . On obtient d’abord des majorations du nombre de résonances dans des petits domaines proches de $2 b q$ . 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 $2 b q$ . 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.

We consider the 3D Schrödinger operator $H = H_{0} + V$ where $H_{0} = {(- i \nabla - A)}^{2} - b$ , $A$ 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 $ℳ$ , and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed Landau level $2 b q$ , $q \in ℕ$ . First, we obtain a sharp upper bound of the number of resonances in a vicinity of $2 b q$ . 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 $2 b q$ . 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.

MR Zbl

DOI : 10.5802/aif.2270

Classification : 35P25, 35J10, 47F05, 81Q10
Mots clés : Magnetic Schrödinger operators, resonances, spectral shift function, Breit-Wigner approximation

Affiliations des auteurs :

Bony, Jean-François ¹ ; Bruneau, Vincent ¹ ; Raikov, Georgi ²

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Bony, Jean-François; Bruneau, Vincent; Raikov, Georgi. Resonances and Spectral Shift Function near the Landau levels. Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 629-671. doi : 10.5802/aif.2270. https://aif.centre-mersenne.org/articles/10.5802/aif.2270/

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