A study of some discrete Dirac equations with principal functions

Abstract

Let L denote the operator generated in ${\ell _{2}( \mathbb{N}, \mathbb{C}^{2}) }$ by

$$\left\{ \begin{array}{l}a_{n+1}y_{n+1}^{( 2) }+b_{n}y_{n}^{(2)}+p_{n}y_{n}^{( 1) }=\lambda y_{n}^{(1)} \\ a_{n-1}y_{n-1}^{(1)} +b_{n}y_{n}^{( 1) }+q_{n}y_{n}^{(2)} =\lambda y_{n}^{(2)},\end{array}\right.\quad n\in \mathbb{N} $$

and the boundary condition

$$(\gamma _{0}+\gamma _{1}\lambda +\gamma _{2}\lambda ^{2})y_{1}^{(2)}+(\beta_{0}+\beta _{1}\lambda +\beta _{2}\lambda ^{2})y_{0}^{(1)}=0$$

where ${( a_{n})}$, ${( b_{n})}$, ${( p_{n}) }$ and ${( q_{n}) }$, ${n\in \mathbb{N} }$ are complex sequences, ${\gamma _{i},\beta _{i} \in \mathbb{C} }$, ${i=0,1,2}$ and ${\lambda }$ is a eigenparameter. With respect to the spectral properties of L, we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities of L, if

$$\sum_{n=1}^{\infty }\exp (\varepsilon n^{\delta }) \big( |1-a_{n}| + |1+b_{n}| + |p_{n}| + |q_{n} | \big) < \infty $$

holds for some ${\varepsilon > 0}$ and ${\delta \in [ \frac{1}{2},1] }$.