03-451 Vincent Bruneau, Alexander Pushnitski, Georgi Raikov

Spectral Shift Function in Strong Magnetic Fields (419K, Postscript) Oct 2, 03

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Abstract. We consider the three-dimensional Schr\"odinger operator $H$ with constant magnetic field of strength $b>0$ and continuous electric potential $V \in L^1({\re^3})$ which admits certain power-like estimates at infinity. We study the asymptotic behaviour as $b \rightarrow \infty$, of the spectral shift function $\xi(E;H,H_0)$ for the pair of operators $(H,H_0)$ at energies $E = {\cal E} b + \lambda$, ${\cal E}>0$ and $\lambda \in \re$ being fixed. We distinguish two asymptotic regimes. In the first one called {\it asymptotics far from the Landau levels} we pick ${\cal E}/2 \not \in {\mathbb Z}_+$ and $\lambda \in \re$; then the main term is always of order $\sqrt{b}$, and is independent of $\lambda$. In the second asymptotic regime called {\it asymptotics near a Landau level} we choose ${\cal E}= 2 q_0$, $q_0 \in {\mathbb Z}_+$, 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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