外电场和磁场中的锯齿形纳米带
Zigzag nanoribbons in external electric and magnetic fields
摘要：
我们考虑了锯齿形纳米带（准一维紧束缚模型）上的薛定谔算符在外加磁场和电势$V$中的作用。磁场垂直于带的平面，电场垂直于纳米颗粒和磁场的轴。如果没有磁场和电场，那么薛定谔（拉普拉斯）算符的谱由两个非平坦带和一个平坦带（具有无限重数的本征值）组成。如果我们打开磁场，那么磁薛定谔算符的谱由一些非平坦带和它们之间的一个平坦带组成。因此，磁场改变了连续谱，但不改变平带。如果我们打开一个弱的电势$V\to 0$，那么有两种情况：（1）平带分裂成小光谱带。我们确定的小领域的谱带的渐近性。(2)未扰动的平带保持平带。我们描述了所有的潜力时，未扰动的平带仍然是平带，当一个分裂成小带的连续谱。此外，我们解决了小电位的逆谱问题。
英文摘要：
We consider the Schrödinger operators on zigzag nanoribbons (quasi-1D tight-binding models) in external magnetic fields and an electric potential $V$. The magnetic field is perpendicular to the plane of the ribbon and the electric field is perpendicular to the axis of the nanoribbon and the magnetic field. If the magnetic and electric fields are absent, then the spectrum of the Schrödinger (Laplace) operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on the magnetic field, then the spectrum of the magnetic Schrödinger operator consists of some non-flat bands and one flat band between them. Thus the magnetic field changes the continuous spectrum but does not the flat band. If we switch on a weak electric potential $V\to 0$, then there are two cases: (1) the flat band splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed flat band remains the flat band. We describe all potentials when the unperturbed flat band remains the flat band and when one splits into the small band of the continuous spectrum. Moreover, we solve inverse spectral problems for small potentials.
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