Spectral shift function in strong magnetic fields, Algebra i Analiz 16 (2004), 207 - 238; see also St

Dedicated to Professor Mikhail Birman on the occasion of his 75th birthday Abstract. The three-dimensional Schrödinger operator H with constant magnetic field of strength b> 0 is considered under the assumption that the electric potential V ∈ L1(R3) admits certain power-like estimates at infinity. The asymptotic behavior as b→ ∞ of the spectral shift function ξ(E;H,H0) is studied for the pair of operators (H, H0) at the energies E = Eb + λ, E> 0 and λ ∈ R being fixed. Two asymptotic regimes are distinguished. In the first regime, called asymptotics far from the Landau levels, we pick E/2 ∈ Z+ and λ ∈ R; then the main term is always of order b, and is independent of λ. In the second asymptotic regime, called asymptotics near a Landau level, we choose E = 2q0, q0 ∈ Z+, and λ = 0; in this case the leading term of the SSF could be of order b or b for different λ. The main object of investigation in the present paper is the spectral shift functio