The spectrum of discrete Dirac operator with a general boundary condition

[摘要] In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space$l_{2}(\mathbb{N},\mathbb{C}^{2})$ by the discrete Dirac system$$ \textstyle\begin{cases} y_{n+1}^{ (2 )} - y_{n}^{ (2 )} + p_{n} y_{n}^{ (1 )} =\lambda y_{n}^{ (1 )},\\ - y_{n}^{ (1 )} + y_{n-1}^{ (1 )} + q_{n} y_{n}^{ (2 )} =\lambda y_{n}^{ (2 )}, \end{cases}\displaystyle \quad n\in \mathbb{N}, $$ and the general boundary condition$$ \sum_{n = 0}^{\infty } h_{n}y_{n} = 0, $$ where λ is a spectral parameter, Δ is the forward difference operator, ( $h_{n}$ ) is a complex vector sequence such that$h_{n} = ( h_{n}^{(1)}, h_{n}^{(2)} )$ , where$h_{n}^{(i)} \in l^{1} ( \mathbb{N} ) \cap l^{2} ( \mathbb{N} )$ ,$i = 1,2$ , and$h_{0}^{(1)} \ne 0$ . 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.

[学科分类] 航空航天科学

[关键词] Eigenparameter;Spectral analysis;Eigenvalues;Spectral singularities;Discrete equation;Dirac equation