Philippe Briet, Georgi Raikov, Eric Soccorsi
Spectral Properties of a Magnetic Quantum Hamiltonian 
on a Strip
(339K, .pdf)

ABSTRACT.  We consider a 2D Schr\"odinger operator $H_0$ with constant 
magnetic field, on a strip of finite width. The spectrum of $H_0$ 
is absolutely continuous, and contains a discrete set of 
thresholds. We perturb $H_0$ by an electric potential $V$ which 
decays in a suitable sense at infinity, and study the spectral 
properties of the perturbed operator $H = H_0 + V$. First, we 
establish a Mourre estimate, and as a corollary prove that the 
singular continuous spectrum of $H$ is empty, and any compact 
subset of the complement of the threshold set may contain at most 
a finite set of eigenvalues of $H$, each of them having a finite 
multiplicity. Next, we introduce the Krein spectral shift function 
(SSF) for the operator pair $(H,H_0)$. We show that this SSF is 
bounded on any compact subset of the complement of the threshold 
set, and is continuous away from the threshold set and the 
eigenvalues of $H$. The main results of the article concern the 
asymptotic behaviour of the SSF at th thresholds, which is 
described in terms of the SSF for a pair of 
effective Hamiltonians.