The main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class \(C_{0}.\) Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Carathéodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established.

中文翻译：

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具有分布势的耗散算子的特征函数的直接方法

本文的主要目的是借助其Cayley变换研究奇异耗散微分算子的光谱特性。证明了耗散微分算子的Cayley变换是具有有限缺陷指数属于\（C_ {0}。\）类的完全非non收缩。利用其特征函数和分辨算子的光谱特性，可以得到耗散微分算子的完整光谱分析。将Cayley变换嵌入其自然的一元整数，即可获得Carathéodory函数。此外，建立了截短的CMV矩阵，该矩阵与耗散微分算子的Cayley变换unit等价。此外，证明了耗散微分算子的逆算子的虚部是秩一算子，并且相关的耗散积分算子的模型算子被构造为半无限三角形矩阵。利用耗散积分算子的特征函数和一阶虚部，建立了相关的Weyl函数。