具有Wigner-von Neumann势的周期Schroedinger算子谱密度的零点
Zeroes of the spectral density of the periodic Schroedinger operator with Wigner-von Neumann potential
摘要：
本文考虑半直线上的Schroedinger算子L_α，背景势为周期势，Wigner-von Neumann势为库仑型：csin（2ωx+d）/（x+1）.我们知道，L_α算符的连续谱具有与自由周期算符相同的带隙结构，而在绝对连续谱的每个带中，存在两个点（所谓临界点或共振点），在这两个点上，L_α算符有一个从属解，它可以是本征值，也可以是“半束缚”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L_α的谱密度在临界点具有幂型零点（即，绝对连续谱具有伪隙）。这种现象在上述意义上是稳定的。
英文摘要：
We consider the Schroedinger operator L_α on the half-line with a periodic background potential and the Wigner-von Neumann potential of Coulomb type: csin(2ωx+d)/(x+1). It is known that the continuous spectrum of the operator L_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 L_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 L_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.
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