Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential
Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential
复制标题
具有维格纳-冯·诺依曼势的周期性薛定谔算子的谱密度零点
DOI：
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发表时间：
2011
影响因子：
0.8
通讯作者：
S. Simonov
中科院分区：
文献类型：
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作者：
S. Naboko;S. Simonov
本文研究了具有周期背景势和库仑型Wigner-von Neumann势csin（2ωx + δ）/（x + 1）的半直线上的Schr dinger算子α.已知算子α的连续谱具有与自由周期算子相同的带隙结构，而在绝对连续谱的每个带中存在两个点（所谓的临界点或共振点），其中算子α具有从属解，该从属解可以是本征值或“半束缚”状态。嵌入本征值现象在边界条件变化和势局部变化的情况下都是不稳定的，也就是说，它不是一般的。我们证明了在一般情况下，算子α的谱密度在临界点处具有幂零（即，绝对连续的频谱具有伪间隙）。这种现象在上述意义上是稳定的。
Abstract We consider the Schrödinger operator α on the half-line with a periodic background potential and the Wigner–von Neumann potential of Coulomb type: csin(2ωx + δ)/(x + 1). It is known that the continuous spectrum of the operator α has the same band-gap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (so-called critical or resonance) where the operator α has a subordinate solution, which can be either an eigenvalue or a “half-bound” state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator α has power-like zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the above-mentioned sense.