Spectral shift function in strong magnetic fields

Bruneau, V.; Pushnitski, A.; Raikov, G.

doi:10.1090/S1061-0022-04-00847-7

Spectral shift function in strong magnetic fields
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by V. Bruneau, A. Pushnitski and G. Raikov

St. Petersburg Math. J. 16 (2005), 181-209

DOI: https://doi.org/10.1090/S1061-0022-04-00847-7

Published electronically: December 17, 2004

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Abstract:

The three-dimensional Schrödinger operator $H$ with constant magnetic field of strength $b>0$ is considered under the assumption that the electric potential $V \in L^1({{\mathbb R}^3})$ admits certain power-like estimates at infinity. The asymptotic behavior as $b \rightarrow \infty$ of the spectral shift function $\xi (E;H,H_0)$ is studied for the pair of operators $(H,H_0)$ at the energies $E = {\mathcal {E}} b + \lambda$, ${\mathcal {E}}>0$ and $\lambda \in {\mathbb R}$ being fixed. Two asymptotic regimes are distinguished. In the first regime, called asymptotics far from the Landau levels, we pick ${\mathcal {E}}/2 \not \in {\mathbb Z}_+$ and $\lambda \in {\mathbb R}$; then the main term is always of order $\sqrt {b}$, and is independent of $\lambda$. In the second asymptotic regime, called asymptotics near a Landau level, we choose ${\mathcal {E}}= 2 q_0$, $q_0 \in {\mathbb Z}_+$, and $\lambda \neq 0$; in this case the leading term of the SSF could be of order $b$ or $\sqrt {b}$ for different $\lambda$.

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