Eigenstructure of nonselfadjoint complex discrete vector Sturm-Liouville problems (Q2570018)

The paper deals with discrete Sturm-Liouville problem \[ \begin{aligned} H(k+1) - \alpha H(k) + \gamma H(k-1) = \lambda H(k), &\quad 1\leq k \leq N-1, \\ F_{s1}H(1) + F_{s2}H(0) = 0, &\quad s=1,\dots,q, \\ L_{s1}H(N) + L_{s2}H(N-1) = 0, &\quad s=1,\dots,q,\end{aligned} \] where \(H(k) \in \mathbb{C}^m,\) \(F_{sj}, s=1,\dots,q, j=1,2,\) are matrices in \(\mathbb{C}^{m \times m}\), not necessarily symmetric, \(\alpha\) and \(\gamma \not= 0\) are complex numbers, and \(\lambda\) is a complex parameter. Eigenvalues and eigenfunctions of (1) - (3) are described. The orthogonal basis in the eigenfunctions space and finite Fourier series expansions in terms of eigenfunctions are studied.