2006/03/31 by J. Bony, J. F. Bony, Vincent Bruneau +6
Mathematics · #35J10 #35P25 #47F05 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35J10 #msc:35P25 #msc:47F05 #msc:81Q10
paper · pdf · doi:10.48550/arxiv.math/0603731
32 pages, 4 figures. Revised version with more precise notation concerning subsets of the Riemann surface
openalex publication_date 2006/03/31 · arxiv created 2006/06/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the 3D Schrödinger operator H = H0 + V where H0 = (-i∇ - A)2, A is a magnetic potential generating a constant magnetic field 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 complex manifold \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 ∈ \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,H0) 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.