论文标题
在可分离的2D几乎Mathieu操作员的光谱上
On the spectra of separable 2D almost Mathieu operators
论文作者
论文摘要
我们考虑可分开的2D离散Schrödinger运算符，该操作员几乎是Mathieu运营商生成的。对于固定的双苯胺频率，我们证明，对于足够小的耦合，频谱必须是一个间隔。这补充了J. Bourgain的结果，确定对于固定耦合，光谱对于某些（积极的）二芬太汀频率具有差距。我们的结果概括为可分离的多维离散schrödinger运算符，由1D准碘二级操作员产生，其潜力是分析性且其频率为二磷酸的。证明是基于对几乎Mathieu操作员光谱厚度的研究，并利用Newhouse Gap Lemma在Cantor套件的总和上。
We consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.
