Vincent Bruneau, Alexander Pushnitski, Georgi Raikov
Spectral Shift Function in Strong Magnetic Fields
(419K, Postscript)
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 $\lambda$.