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Jean-François Bony; Vincent Bruneau; Georgi Raikov
Resonances and Spectral Shift Function near the Landau levels
(Résonances et fonction de décalage spectral près des niveaux de Landau)
Annales de l'institut Fourier, 57 no. 2 (2007), p. 629-671, doi: 10.5802/aif.2270
Article PDF | Reviews MR 2310953 | Zbl 1129.35053
Class. Math.: 35P25, 35J10, 47F05, 81Q10
Keywords: Magnetic Schrödinger operators, resonances, spectral shift function, Breit-Wigner approximation

Résumé - Abstract

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 ${\mathcal{M}}$, 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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