论文标题
在层次结构Schrödinger类型运算符上
On the spectrum of the hierarchical Schrödinger type operators
论文作者
论文摘要
本文的目的是对schrödinger类型运算符的光谱分析$ h = l+v $，taibleson-vladimirov乘数的扰动$ l = \ mathfrak {d}^α$ a潜在的$ v $。假设$ v $属于某些类别的电位，我们表明，$ h $的频谱的离散部分可能包含负能量，它也出现在$ l $的频谱差距中。我们将在两个部分中将$ h $的频谱分成：高能量部分，其中包含特征值，这些特征值对应于潜在的$ V，$和低能部分的特征函数，该函数位于某些有界的Schrödinger-type操作员上，该操作员在Dyson等级Lattice上演奏。我们特别注意稀疏电位的类别。在这种情况下，我们获得了$ h $的精确光谱渐进性，只要位置之间的距离顺序倾向于足够快地无限。我们还获得了有关$ h $的本地化理论的某些结果，但要受（非共性）随机潜在$ v $。例子说明了我们的方法。
The goal of this paper is the spectral analysis of the Schrödinger type operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^α$ by a potential $V$. Assuming that $V$ belongs to a certain class of potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V,$ and low energy part which lies in the spectrum of certain bounded Schrödinger-type operator acting on the Dyson hierarchical lattice. We pay special attention to the class of sparse potentials. In this case we obtain precise spectral asymptotics for $H$ provided the sequence of distances between locations tends to infinity fast enough. We also obtain certain results concerning localization theory for $H$ subject to (non-ergodic) random potential $V$. Examples illustrate our approach.
