Resonances and spectral shift function near the Landau levels
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Date
2007
Journal Title
Journal ISSN
Volume Title
Publisher
ANNALES DE L INSTITUT FOURIER
Abstract
We consider the 3D Schrodinger operator H = H-0 + V where H-0 = (-i del - 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 is an element of 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.
Description
Keywords
magnetic Schrodinger operators, resonances, spectral shift function, Breit-Wigner approximation, MAGNETIC SCHRODINGER-OPERATORS, TRACE FORMULA, HYDROGEN-ATOM, PERTURBATIONS, ASYMPTOTICS, FIELDS