Principal Functions of Nonselfadjoint Discrete Dirac Equations with Spectral Parameter in Boundary Conditions

Abstract

Let $L$ denote the operator generated in ${\ell }_2(\mathbb{N},{ℂ}^2)$ by $a_{n+1}{y}_{n+1}^{(2)}+b_n{y}_n^{(2)}+p_n{y}_n^{(1)}=\lambda y_n^{(1)}$, $a_{n-1}{y}_{n-1}^{(1)}+b_n{y}_n^{(1)}+q_n{y}_n^{(2)}=\lambda y_n^{(2)}$, $n\in \mathbb{N}$, and the boundary condition $({\gamma }_0+{\gamma }_1\lambda )y_1^{(2)}+({\beta }_0+{\beta }_1\lambda )y_0^{(1)}=\mathbf{0}$, where $(a_n)$, $(b_n)$, $(p_n)$, and $(q_n)$, $n\in \mathbb{N}$ are complex sequences, ${\gamma }_i,{\beta }_i\in ℂ$, $i=\mathrm{0,1}$, and $\lambda$ is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of $L$.