We investigate the self-adjointness of the two-dimensional Dirac operator $D$, with $quantum$-$dot$ and $Lorentz$-$scalar$ $\delta$-$shell$ boundary conditions, on piecewise $C^2$ domains (with finitely many corners). For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space $H^{1/2}$, the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of the adjoint operator $D^*$ in terms of the domain of $D$ and the set of harmonic functions that verify some $mixed$ boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.

中文翻译：

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角域上二维狄拉克算子的自伴随性

我们研究了二维狄拉克算子 $D$ 的自伴随性，具有 $quantum$-$dot$ 和 $Lorentz$-$scalar$ $\delta$-$shell$ 边界条件，在分段 $C^2 上$ 域（具有有限多个角）。对于这两个模型，我们证明了独特的自伴随实现的存在，其域包含在 Sobolev 空间 $H^{1/2}$ 中，即自由狄拉克算子的形式域。我们论文的主要部分包括根据 $D$ 的域和验证某些 $mixed$ 边界条件的调和函数集来描述伴随算子 $D^*$ 的域。然后，我们详细研究了无限扇区上的问题，其中可以进行显式计算：我们找到了这种情况的自伴随扩展。然后通过坐标变换将结果转换为一般域。