Spectral properties of Sturm-Liouville system with eigenvalue-dependent boundary conditions

In this paper, we consider the boundary value problem y(j)'' + lambda(2)y(j) = Sigma(n)(k=1) V-jk(x)y(k,) x is an element of R+ := (0,infinity) y(j)''(0) +(alpha(0)+alpha(1)lambda+alpha(2)lambda(2))y(j)(0) = 0, j = 1,2, .... , n, where lambda is the spectral parameter V(x) = vertical bar vertical bar V-jk(x)vertical bar vertical bar(n)(1) and is a Hermitian matrix such that integral(infinity)(x) t vertical bar V(t)vertical bar dt < infinity, x is an element of R+ and alpha(i) is an element of c,i =0,1,2 with alpha(2) not equal 0 In this paper, we investigate the eigenvalues and spectral singularities of L. In particular, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, under the Naimark and Pavlov conditions.