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 $\lambda$.