Abstract

We study the problem of the electron excitation spectrum in the presence of point-like and regularized Coulomb impurities in gapped graphene. To this end, we use the Dirac model and in the point-like case theory of self-adjoint extensions of symmetric operators. In the point-like case, we construct a family of self-adjoint Hamiltonians describing the excitations for any charge of an impurity. Spectra and (generalized) eigenfunctions for all such Hamiltonians are found. Then, we consider the spectral problem in the case of a regularized Coulomb potential of impurities for a special regularization. We study exact equations for charges of impurities that may generate bound states with energy that coincides with the upper boundary of the negative branch of the continuous spectrum (supercritical charges) and calculate these charges.

中文翻译：

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接近库仑杂质的石墨烯中电子激发的光谱

摘要

我们研究了在间隙石墨烯中存在点状和规则化库仑杂质时电子激发光谱的问题。为此，我们使用Dirac模型和对称算子的自伴随扩展的点状案例理论。在点状情况下，我们构造了一个自伴随的哈密顿量族，描述了任何杂质电荷的激发。找到所有这些哈密顿量的谱和（广义）本征函数。然后，在特殊的正则化杂质的正则库仑势的情况下，我们考虑光谱问题。我们研究了杂质电荷的精确方程式，这些电荷可能会生成束缚态，其能量与连续光谱的负分支的上限（超临界电荷）一致，并计算出这些电荷。