Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials.
Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials.
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施诺尔定理和具有标量势的无质量狄拉克算子的谱特性。
DOI：
10.1007/s11005-015-0799-1
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Tomio Umeda
中科院分区：
文献类型：
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作者：
Karl Michael Schmidt;Tomio Umeda
无质量狄拉克算子的谱是人们非常感兴趣的，例如，对于石墨烯的电子性质，但光谱间隙的存在等基本问题仍然是开放的。我们表明，适当的实值潜力的无质量狄拉克运营商的特征值位于小的集合内容易表征的潜在的性质，我们证明了Schnol型定理有关谱点的多项式有界性的狄拉克方程的解决方案。此外，我们表明，在最低限度的假设下，离开基本上不受限制的潜力在大部分的空间，频谱的无质量狄拉克运营商涵盖了整个真实的线，特别是，这将是这种情况下，如果潜在的是几乎恒定的一系列区域。
The spectra of massless Dirac operators are of essential interest, e.g., for the electronic properties of graphene, but fundamental questions such as the existence of spectral gaps remain open. We show that the eigenvalues of massless Dirac operators with suitable real-valued potentials lie inside small sets easily characterized in terms of properties of the potentials, and we prove a Schnol-type theorem relating spectral points to polynomial boundedness of solutions of the Dirac equation. Moreover, we show that, under minimal hypotheses which leave the potential essentially unrestrained in large parts of space, the spectrum of the massless Dirac operator covers the whole real line; in particular, this will be the case if the potential is nearly constant in a sequence of regions.