Resonances and Spectral Shift Function near the Landau levels

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

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

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 $2bq$, $q\in \mathbb{N}$. 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.

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 $2bq$, $q\in \mathbb{N}$. 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.

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