On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

Abstract

We consider the three-dimensional Schrodinger operators \( H_0 \) and \( H_\pm \) where \( H_{0} = (i\nabla + A)^{2} - b \) , A is a magnetic potential generating a constant magnetic field of strength \( b > 0 \) , and \( H_{\pm} = H_{0} \pm V \) where \( b \geq 0 \) decays fast enough at infinity. Then, A. Pushnitski’s representation of the spectral shift function (SSF) for the pair of operators \( H_\pm, H_0 \) is well defined for energies \( E \neq 2qb,\, q \in \mathbb{Z}_{+}. \) We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels \( 2qb,\, q \in \mathbb{Z}_{+}. \) Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity.