缓变磁场引起的低能隙
Low lying spectral gaps induced by slowly varying magnetic fields
摘要：
我们考虑一个二维周期薛定谔算子，它被一个弱磁场扰动，弱磁场的强度在正平均值附近缓慢变化。我们在很大程度上表明，相应的磁薛定谔算子的频谱的底部开发的光谱岛分开的差距，提醒朗道水平结构。首先，我们构造了一个有效的Hofstadter磁矩阵，它精确地描述了全算子的低频谱。这种有效磁矩阵的构造不需要非磁算符的谱中存在间隙，只需要第一和第二布洛赫本征值不交叉，但它们的范围可能重叠。交叉情况更加困难，将在其他地方考虑。其次，我们进行了详细的谱分析的有效矩阵使用规范协变磁伪微分学适应缓慢变化的磁场。作为一个应用程序，我们证明了重叠的情况下，分离的间隙谱岛的外观。
英文摘要：
We consider a periodic Schrödinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schrödinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.
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