On the structure of discrete spectrum of a non-selfadjoint system of differential equations with integral boundary condition

In this paper, we investigated the spectrum of the operator generated in Hilbert Space of vector-valued functions L-2(R+, C-2) by the system 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 is an element of R+ := (0, infinity), and the integral boundary condition of the type integral(infinity)(0) K(x,t)y(t,lambda)dt + alpha y(2)(0,lambda) - beta y(1)(0, lambda) = 0 where. is a complex parameter, q(i), i = 1, 2 are complex-valued functions and alpha, beta is an element of C. K(x,t) is vector fuction such that K(x, t) = (K-1(x, t), K-2(x, t)), K-i (x, t) is an element of L-1(0, infinity) boolean AND L-2(0, lambda), i = 1, 2. Under the condition vertical bar qi (x)vertical bar <= ce(-epsilon) (root x), c > 0, epsilon > 0, i = 1, 2 we proved that L(lambda) has a finite number of eigenvalues and spectral singularities with finite multiplicities.