03-452 Claudio Fernandez, Georgi Raikov
On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels (273K, Postscript) Oct 3, 03
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Abstract. We consider the three-dimensional Schr\"odinger 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 $V \geq 0$ 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.

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