扶手椅型纳米管上的磁性薛定谔算符
Magnetic Schrödinger operators on armchair nanotubes
摘要：
我们考虑具有周期势的薛定谔算子在$\R^3$中的扶手椅型纳米管的准一维连续周期模型上，该模型在平行于纳米管轴的均匀磁场（振幅为$B\in \R$）中。该算子的谱由一个绝对连续的部分（由间隙分隔的谱带）加上具有无限重数的无穷多个本征值组成。我们描述了所有具有相同特征值的特征函数，包括紧支集。我们将光谱描述为$B$的函数。对于某些特定的势，我们证明了独立于磁场的间隙的存在。如果$B\ne 0$，则存在无限个长度为$|G_n|\to\iy$的能隙$G_n$为$n\to\iy$，并且我们确定了固定$B$的能隙在高能时的渐近性。此外，我们确定的间隙$G_n$为$B\to 0$固定$n$的渐近。
英文摘要：
We consider the Schrödinger operator with a periodic potential on a quasi 1D continuous periodic model of armchair nanotubes in $\R^3$ in a uniform magnetic field (with amplitude $B\in \R$), which is parallel to the axis of the nanotube. The spectrum of this operator consists of an absolutely continuous part (spectral bands separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all eigenfunctions with the same eigenvalue including compactly supported. We describe the spectrum as a function of $B$. For some specific potentials we prove an existence of gaps independent on the magnetic field. If $B\ne 0$, then there exists an infinite number of gaps $G_n$ with the length $|G_n|\to\iy$ as $n\to\iy$, and we determine the asymptotics of the gaps at high energy for fixed $B$. Moreover, we determine the asymptotics of the gaps $G_n$ as $B\to 0$ for fixed $n$.
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