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

In this paper, we consider the boundary value problem $$y^{\prime\prime}_{j} + \lambda^{2}y_{j} = \sum^{n}_{k = 1} V_{jk} (x) y_{k}, \quad x \in{\mathbb R}_{+} : = ( 0, \infty),$$$$y^{\prime}_{j} (0) + (\alpha_{0} + \alpha_{1} \lambda+ \alpha_{2} \lambda^{2}) y_{j} (0) = 0, \quad j = 1, 2, \ldots, n,$$ where λ is the spectral parameter and $V (x) = \Vert V_{jk} (x)\Vert^{n}_{1}$ is a Hermitian matrix such that $$\int^{\infty}_{x} t\vert V (t)\vert dt < \infty, \quad x \in {\mathbb R}_{+}$$ and αi ∈ ℂ, i = 0, 1, 2, with α2 ≠ 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.