The spectrum of discrete Dirac operator with a general boundary condition

Abstract

AbstractIn this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space $l_{2}(\mathbb{N},\mathbb{C}^{2})$ l 2 ( N , C 2 ) by the discrete Dirac system $$ \textstyle\begin{cases} y_{n+1}^{ (2 )} - y_{n}^{ (2 )} + p_{n} y_{n}^{ (1 )} =\lambda y_{n}^{ (1 )},\\ - y_{n}^{ (1 )} + y_{n-1}^{ (1 )} + q_{n} y_{n}^{ (2 )} =\lambda y_{n}^{ (2 )}, \end{cases}\displaystyle \quad n\in \mathbb{N}, $$ { y n + 1 ( 2 ) − y n ( 2 ) + p n y n ( 1 ) = λ y n ( 1 ) , − y n ( 1 ) + y n − 1 ( 1 ) + q n y n ( 2 ) = λ y n ( 2 ) , n ∈ N , and the general boundary condition $$ \sum_{n = 0}^{\infty } h_{n}y_{n} = 0, $$ ∑ n = 0 ∞ h n y n = 0 , where λ is a spectral parameter, Δ is the forward difference operator, ($h_{n}$ h n ) is a complex vector sequence such that $h_{n} = ( h_{n}^{(1)}, h_{n}^{(2)} )$ h n = ( h n ( 1 ) , h n ( 2 ) ) , where $h_{n}^{(i)} \in l^{1} ( \mathbb{N} ) \cap l^{2} ( \mathbb{N} )$ h n ( i ) ∈ l 1 ( N ) ∩ l 2 ( N ) , $i = 1,2$ i = 1 , 2 , and $h_{0}^{(1)} \ne 0$ h 0 ( 1 ) ≠ 0 . Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity.