Spectral properties of a magnetic quantum Hamiltonian on a strip
Spectral properties of a magnetic quantum Hamiltonian on a strip
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带状磁量子哈密顿量的光谱特性
作者：
P. Briet;G. Raikov;É. Soccorsi
本文考虑有限宽带上的二维常磁场薛定谔算子H_0。H0的谱是绝对连续的，并且包含一组离散的阈值。我们用一个在无穷远处在适当意义下衰减的电势V扰动H_0，研究扰动算子H = H_0 + V的谱性质.首先，我们建立了一个新的估计，作为推论，证明了H的奇异连续谱是空的，并且阈值集的补的任何紧子集至多包含H的一个有限特征值集，它们中的每一个都具有有限的多重性。其次，我们引入了算子对（H，H0）的Krein谱位移函数（SSF）。我们证明了这个SSF在阈值集的补的任何紧子集上是有界的，并且是远离阈值集和H的特征值连续的。本文的主要结果是关于在阈值处的SSF的渐近行为，它是用一对相互作用的Hamilton算子的SSF来描述的。
We consider a 2D Schrodinger operator H0 with constant magnetic field, on a strip of finite width. The spectrum of H0 is absolutely continuous, and contains a discrete set of thresholds. We perturb H0 by an electric potential V which decays in a suitable sense at infinity, and study the spectral properties of the perturbed operator H = H0 + 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 intro- duce the Krein spectral shift function (SSF) for the operator pair (H,H0). 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 eective Hamiltonians.