LAUSR

In this paper, we investigated the spectrum of the operator $$L(\lambda )$$L(λ) generated in Hilbert Space of vector-valued functions $$L_{2}(\mathbb {R}_{+},C_{2})$$L2(R+,C2) by the system $$\begin{aligned} iy_{1}^{'}(x,\lambda )+q_{1}(x)y_{2}(x,\lambda )&=\lambda y_{1}(x,\lambda )\\ -iy_{2}^{'}(x,\lambda )+q_{2}(x)y_{1}(x,\lambda )&=\lambda y_{2}(x,\lambda ),x\in \mathbb {R}_{+}:=(0,\infty ), \end{aligned}$$iy1′(x,λ)+q1(x)y2(x,λ)=λy1(x,λ)-iy2′(x,λ)+q2(x)y1(x,λ)=λy2(x,λ),x∈R+:=(0,∞),and the integral boundary condition of the type $$\begin{aligned} \int _{0}^{\infty }K(x,t)y(t,\lambda ){\mathrm {dt}}+\alpha y_{2}(0,\lambda )-\beta y_{1}(0,\lambda )=0 \end{aligned}$$∫0∞K(x,t)y(t,λ)dt+αy2(0,λ)-βy1(0,λ)=0where $$\lambda $$λ is a complex parameter, $$q_{i},\,i=1,2$$qi,i=1,2 are complex-valued functions and $$\alpha ,\beta \in \mathbb {C}$$α,β∈C. K(x, t) is vector fuction such that $$K(x,t)=(K_{1}(x,t),K_{2}(x,t)),\,K_{i}(x,t)\in L_{1}(0,\infty )\cap L_{2}(0,\lambda ),\,i=1,2$$K(x,t)=(K1(x,t),K2(x,t)),Ki(x,t)∈L1(0,∞)∩L2(0,λ),i=1,2. Under the condition $$\begin{aligned} \left| q_{i}(x)\right| \le ce^{-\varepsilon \sqrt{x}},c>0,\varepsilon >0,i=1,2 \end{aligned}$$qi(x)≤ce-εx,c>0,ε>0,i=1,2we proved that $$L(\lambda )$$L(λ) has a finite number of eigenvalues and spectral singularities with finite multiplicities.