无界单调拟周期算子扰动级数的收敛性
Convergence of perturbation series for unbounded monotone quasiperiodic operators
摘要：
我们考虑一类无界的拟周期薛定谔型算子$\ell^2(\mathbb Z^d)$与单调的潜力（类似于马里兰州的模型），并表明，瑞利-薛定谔扰动系列这些运营商收敛在小动能的制度，一致的频谱。作为结果，我们得到了一个新的证明，安德森局部化在一个更一般的比以前的一类这样的运营商，明确的收敛级数展开的特征值和特征向量。这个结果可以被限制到一个能量窗口，如果潜在的只是局部单调和一对一。这种方法的修改也允许潜在的非严格单调，并有一个平坦的部分，在额外的限制下的频率。
英文摘要：
We consider a class of unbounded quasiperiodic Schrödinger-type operators on $\ell^2(\mathbb Z^d)$ with monotone potentials (akin to the Maryland model) and show that the Rayleigh--Schrödinger perturbation series for these operators converges in the regime of small kinetic energies, uniformly in the spectrum. As a consequence, we obtain a new proof of Anderson localization in a more general than before class of such operators, with explicit convergent series expansions for eigenvalues and eigenvectors. This result can be restricted to an energy window if the potential is only locally monotone and one-to-one. A modification of this approach also allows the potential to be non-strictly monotone and have a flat segment, under additional restrictions on the frequency.
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