Spectrum of the Sturm-Liouville operators with boundary conditions polynomially dependent on the spectral parameter

In this paper, we consider the operator L generated in L-2(R+) by the Sturm-Liouville equation -y '' + q(x)y = lambda(2)y, chi is an element of R+ = [0,infinity), and the boundary condition (alpha(0) + alpha(1)lambda + alpha(2)(2 lambda))y' (0) - (beta(0) + beta(1)lambda + beta(2)lambda(2))y(0) = 0, where q is a complex-valued function, alpha(i), beta(i) is an element of C, i = 0, 1, 2, and lambda is an eigenparameter. Under the conditions q, q' is an element of AC((R)+), lim(x ->infinity) vertical bar q(x)vertical bar + vertical bar q'(x)vertical bar = 0, sup(chi is an element of R+) [e(epsilon)root(chi)vertical bar q ''(chi)vertical bar] < infinity, epsilon > 0, using the uniqueness theorems of analytic functions, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities.