Abstract

The Cauchy problem with a localized initial condition for the two-dimensional massless Dirac equation (describing quantum states in graphene [1]) with linear potential is considered. Applying the \(h\)-Fourier transform with respect to the spatial variables to a solution \(\Psi\), we can write, in some domains, a decomposition of \(\widetilde \Psi(p, t)\) of the form \(\widetilde \Psi = \sum_{\sigma \in \pm} e^{i S^\pm / h} \eta^\pm \) (an expansion in the modes \(\eta^\pm \)). The specific feature of this problem is that the phases and modes of this expansion have different form depending on the quantity \(p_2/h^\frac12\). For \(p_2 \gg h^\frac12 \), \(\widetilde \Psi\) is constructed in the standard WKB form, where \(\eta^\pm\) is constructed in the form of a series in powers of \(h\). For \(|p_2| \ll h^\frac12 \), \(\widetilde \Psi\) is constructed by the Kucherenko method, using the Duhamel principle.

DOI 10.1134/S1061920821020126

中文翻译：

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具有线性势和局部初始条件的二维无质量狄拉克方程中特征交集的影响

摘要

考虑了具有线性势的二维无质量狄拉克方程（描述石墨烯 [1] 中的量子态）的局部初始条件的柯西问题。将关于空间变量的\(h\) -傅立叶变换应用于解决方案\(\Psi\)，我们可以在某些域中编写\(\widetilde \Psi(p, t)\)的分解形式\(\widetilde \Psi = \sum_{\sigma \in \pm} e^{i S^\pm / h} \eta^\pm \)（模式\(\eta^\下午\) )。这个问题的具体特征是这个膨胀的阶段和模式根据数量\(p_2/h^\frac12\)具有不同的形式。对于\(p_2 \gg h^\frac12 \)，\(\widetilde \Psi\)以标准 WKB 形式构造，其中\(\eta^\pm\)以\(h\) 的幂级数形式构造。对于\(|p_2| \ll h^\frac12 \)，\(\widetilde \Psi\)由 Kucherenko 方法构造，使用 Duhamel 原理。

DOI 10.1134/S1061920821020126