2012/05/15 by J. -F. Bony, Bony, J. -F., V. Bruneau +3
Mathematics · Physics and Astronomy · #35J10 #35P25 #47F05 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.AP #math.MP #math.SP #msc:35J10 #msc:35P25 #msc:47F05 #msc:81Q10
paper · pdf · doi:10.48550/arxiv.1205.3362
22 pages, 2 figures, typos corrected; to appear in the proceedings of the conference Spectral and Scattering Theory and Related Topics, RIMS, Kyoto, Japan, 14 - 17 December 2011, RIMS Kokyuroku Bessatsu
arxiv created 2013/04/26 · arxiv updated 2013/04/29
In this survey article we consider the operator pair (H,H0) where H0 is the shifted 3D Schrödinger operator with constant magnetic field, H : = H0 + V, and V is a short-range electric potential of a fixed sign. We describe the asymptotic behavior of the Krein spectral shift function (SSF) ξ(E; H,H0) as the energy E approaches the Landau levels 2bq, q ∈ \mathbb Z+, which play the role of thresholds in the spectrum of H0. The main asymptotic term of ξ(E; H,H0) as E → 2bq with a fixed q ∈ \mathbb Z+ is written in the terms of appropriate compact Berezin-Toeplitz operators. Further, we investigate the relation between the threshold singularities of the SSF and the accumulation of resonances at the Landau levels. We establish the existence of resonance free sectors adjoining any given Landau level and prove that the number of the resonances in the complementary sectors is infinite. Finally, we obtain the main asymptotic term of the local resonance counting function near an arbitrary fixed Landau level; this main asymptotic term is again expressed via the Berezin-Toeplitz operators which govern the asymptotics of the SSF at the Landau levels.