Convergence of perturbation series for unbounded monotone quasiperiodic operators
Convergence of perturbation series for unbounded monotone quasiperiodic operators
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DOI：
10.1016/j.aim.2022.108647
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发表时间：
2020-05
影响因子：
1.7
通讯作者：
I. Kachkovskiy;L. Parnovski;R. Shterenberg
中科院分区：
文献类型：
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作者：
I. Kachkovskiy;L. Parnovski;R. Shterenberg
我们考虑一类在 ℓ 2 (Z d) 上具有单调势（类似于马里兰模型）的无界准周期薛定谔型算子，并表明这些算子的瑞利-薛定谔扰动级数在小动能范围内收敛，在谱中均匀。因此，我们在比以前更通用的此类算子中获得了安德森定位的新证明，并具有特征值和特征向量的显式收敛级数展开。如果势仅是局部单调和一对一的，则该结果可以限制在能量窗口内。这种方法的修改还允许在对频率的额外限制下实现非严格单调并具有平坦段。
We consider a class of unbounded quasiperiodic Schrödinger-type operators on ℓ 2 (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 frequencies.