Spectral properties of the finite system of Klein-Gordon S-wave equations with condition depends on spectral parameter

The spectral characteristics of the operator $L$ are studied where $L$ is defined within the Hilbert space $L_{2}(\mathbb{R}_{+}, \mathbb{C}^{V})$ given by a finite system of Klein-Gordon type differential equations and boundary condition depends on spectral parameter. The research of the Klein-Gordon type operator continues to be an important topic for researchers due to the range of applicability of them in numerous branches of mathematics and quantum physics. Contrary to the previous works, we take the potential as complex valued and generalize the problem to the matrix Klein-Gordon operator case. The spectrum is derived by determining the Jost function and resolvent operator of the prescribed operator. Further, we provide the conditions that must be met for the certain quantitative properties of the spectrum.