The integrated density of states and its absolute continuity for magnetic Schr\"odinger operators with unbounded random potentials
The integrated density of states and its absolute continuity for magnetic Schr\"odinger operators with unbounded random potentials
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无界随机势磁薛定谔算子的积分态密度及其绝对连续性
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
S. Warzel
中科院分区：
文献类型：
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作者：
T. Hupfer;H. Leschke;P. Müller;S. Warzel
本文的研究对象是多维欧几里得空间中量子粒子的态的积分密度，该空间以一个Schrödinger算子为特征，具有磁场和无界随机势。在恒定磁场和遍历随机势的情况下，我们证明了态的积分密度作为有限体积算子的合适空间特征值集中的无限体积极限的存在性，以及它与所选边界条件的独立性和它的几乎肯定的非随机性。此外，状态的积分密度用无限体积Schrödinger算子的空间局域谱族表示。最后，对相当一般的磁场和某些允许所谓的单参数分解的随机电位，导出了Wegner估计。该估计暗示了状态密度积分的绝对连续性，并给出了其导数状态密度的显式上界。此外，我们还给出了具有Neumann边界条件的Schrödinger算子的抗磁性不等式。
The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schrödinger operator with magnetic field and unbounded random potential. In case of a constant magnetic field and an ergodic random potential, we prove the existence of the integrated density of states as the infinite-volume limit of suitable spatial eigenvalue concentrations of finite-volume operators as well as its independence of the chosen boundary conditions and its almost-sure nonrandomness. Moreover, the integrated density of states is expressed in terms of the spatially localized spectral family of the infinite-volume Schrödinger operator. Finally, a Wegner estimate is derived for rather general magnetic fields and certain random potentials admitting a so-called one-parameter decomposition. The estimate implies the absolute continuity of the integrated density of states and provides explicit upper bounds on its derivative, the density of states. Besides we show a diamagnetic inequality for Schrödinger operators with Neumann boundary conditions.