 07166 Philippe Briet, Georgi Raikov, Eric Soccorsi
 Spectral Properties of a Magnetic Quantum Hamiltonian
on a Strip
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Jun 29, 07

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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.
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