发表机构
Technische Universität Graz; Czech Technical University in Prague; Technische Universität Dortmund（格拉茨工业大学; 布拉格捷克理工大学; 多特蒙德工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究支撑于折线局部形变曲线上的洛伦兹标量δ壳相互作用的狄拉克算子，证明其能隙内离散本征值有限，且当τ足够小或绝对值足够大时离散谱非空，该谱由几何结构诱导。
AI 中文摘要
我们考虑平面内带有正质量的有质量狄拉克算子，其支撑于C^∞光滑曲线Σ⊂ℝ²上的洛伦兹标量δ壳相互作用具有强度τ∈(-∞,0)\{-2}，该曲线是折线的局部形变。该奇异相互作用通过在算子定义域内的曲线Σ上施加合适的传输条件来定义。此类狄拉克算子是自伴的，且在本质谱中存在一个能隙，其大小是显式的，取决于质量和相互作用强度。我们证明该能隙内的离散本征值数目是有限的。在Σ所界定的某个区域为凸的假设下，我们证明只要τ足够小或绝对值足够大，对应的狄拉克算子具有非空的离散谱。该结果适用于上述任意开口角的折线扰动，且该离散谱由几何结构诱导，因为对于支撑于直线上的同类型奇异相互作用，离散谱为空。
英文摘要
We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $δ$-shell interaction of strength $τ\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $Σ\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $Σ$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $Σ$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $τ$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.
Comments29 pages; comments welcome!
