二维准周期势的动量空间多尺度分析
Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two
摘要：
我们考虑二维多调和算子$H=(-Δ)^l+V(\x)$，其中$l\geq 2$，$l$为整数，且具有准周期势$V(\x)$ .我们证明了$H$的绝对连续谱包含一个半轴，并且在这个半轴的每一点上都存在一族广义本征函数，它们具有以下性质.首先，在高能区本征函数接近于平面波$e^{i< \k,\x>}$。第二，对应于这些本征函数的动量空间$\k$中的等能曲线具有轻微扭曲的带孔圆的形式（Cantor型结构）。发展了一种新的动量空间多尺度分析方法来证明这些结果。
英文摘要：
We consider a polyharmonic operator $H=(-Δ)^l+V(\x)$ in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential $V(\x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i< \k,\x>}$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.
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