Schrödinger Operators on Zigzag Nanotubes-猫眼课题宝

之字形纳米管上的薛定谔算子

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2007

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作者：

E. Korotyaev;I. Lobanov

摘要。我们考虑了具有周期势的Schrödinger算子在之字形单壁碳纳米管准一维模型上的作用。该算子的谱由一个绝对连续的部分（间隔分隔的区间）加上无穷多个具有无穷多重性的特征值组成。我们描述了具有相同特征值的所有紧支持特征函数。我们定义了一个李雅普诺夫函数，它在黎曼曲面上是解析的。在每一张纸上，李雅普诺夫函数都有和标量情况下相同的性质，但是它有分支点，我们称之为共振。我们证明了所有共振都是实的。我们确定了周期谱和反周期谱以及高能量共振的渐近性。我们证明存在两种类型的间隙：i)稳定间隙，其中端点是周期和反周期特征值，ii)不稳定（共振）间隙，其中端点是共振（即Lyapunov函数的实分支点）。我们描述了所有的有限间隙势。我们展示了映射：势能

Abstract.We consider the Schrödinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials. We show that the mapping: potential $$ ightarrow$$ all eigenvalues is a real analytic isomorphism for some class of potentials.

Schrödinger Operators on Zigzag Nanotubes

之字形纳米管上的薛定谔算子

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