07-166 Philippe Briet, Georgi Raikov, Eric Soccorsi

Spectral Properties of a Magnetic Quantum Hamiltonian on a Strip (339K, .pdf) 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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