AI 中文总结
本文研究带壳相互作用的二维朗道-狄拉克算子，分析其自伴性，证明非临界情形下本征值随曲线对数容量在朗道-狄拉克能级处积累，还得到外边界值问题的本征值聚类结果。
AI 中文摘要
我们考虑二维狄拉克算子，其受恒定磁场扰动，扰动项为静电与洛伦兹标量δ相互作用的组合，该相互作用具有变系数且支撑于一条光滑闭曲线上。我们在所谓非临界与临界情形下研究其自伴性：非临界情形中，本质谱保持不变，仍为朗道-狄拉克能级集合（即未受扰算子的无穷重本征值）；临界情形中，含零的能隙内会出现额外的本质谱区间。我们的主要结果涉及非临界情形下的离散谱：利用相关边界积分算子的伪微分性质，我们证明本征值以曲线的对数容量决定的速率在每个朗道-狄拉克能级处积累；一个新颖且令人惊讶的现象是，积累的一侧会随相对于临界值的位置变化而改变。作为副产品，通过束缚耦合得到了一族外边界值问题的本征值聚类，无穷质量边界条件是其特例。
英文摘要
We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so called non critical and critical cases. In the non-critical case the essential spectrum is unchanged - it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator - while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.
