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.

我们考虑由一维近乎 Mathieu 算子生成的可分离二维离散薛定谔算子。对于固定的丢番图频率，我们证明对于足够小的耦合，频谱必须是一个区间。这补充了 J. Bourgain 确定的结果，即对于固定耦合，频谱对于某些（正测量）丢番图频率具有间隙。我们的结果推广到由一维准周期算子生成的可分离多维离散薛定谔算子，其潜力是解析的，频率是丢番图。该证明基于对几乎 Mathieu 算子的频谱厚度的研究，并在 Cantor 集的和上使用 Newhouse Gap Lemma。