AI 中文总结
本文在特定Hilbert空间中研究耗散奇异q-Dirac算子，通过构造自伴膨胀等方法得到其散射矩阵、特征函数，并证明相关向量系的完备性。
AI 中文摘要
本文在Hilbert空间$\u2112_{q}^{2}(q^{\u2124};\u2102^{2})$中研究耗散奇异$q$-Dirac算子，这类算子在极限点情形下作为极小对称算子的扩张出现。我们构造自伴膨胀并得到其入射和出射谱表示，由此可显式计算与该膨胀相关的散射矩阵。此外，我们为耗散算子构建泛函模型，并利用对应自伴算子的Titchmarsh-Weyl函数表示其特征函数。最后，我们得到这类耗散Dirac算子的本征向量及相关向量系的完备性结果。
英文摘要
In this paper, dissipative singular $q$-Dirac operators are examined in Hilbert space $\mathcal{L}_{q}^{2}(q^{\mathbb{Z}};\mathbb{C}^{2})$, where they arise as extensions of a minimal symmetric operator in the limit-point case. We develop a selfadjoint dilation and get its incoming and outgoing spectral representations, enabling the explicit computation of the scattering matrix associated with the dilation. Furthermore, we construct a functional model for the dissipative operator and express its characteristic function in terms of the Titchmarsh-Weyl function of the corresponding selfadjoint operator. Finally, we establish results concerning the completeness of the system of eigenvectors and associated vectors for these dissipative Dirac operators.
