The cubic eigenparameter dependent discrete Dirac equations with principal functions

Abstract

Summary: Let us consider the boundary value problem (BVP) for the discrete Dirac equations) \[\begin{cases} a_{n+1}y_{n+ 1}^{(2)}+ b_ny^{(2)}_n+ p_n y^{(1)}_n & =\lambda y^{(1)}_n,\\ a_{n-1}y^{(1)}_{n-1}+ b_n y_n^{(1)}+q_n y_n^{(2)}& =\lambda y_n^{(2)},\qquad n\in\mathbb{N},\end{cases}\tag{\(0.1\)}\] \[(\gamma_0+ \gamma_1\lambda+ \gamma_2 \lambda^2+ \gamma_3\lambda^3) y^{(2)}_1+ (\beta_0+ \beta_1 \lambda+ \beta_2\lambda^2) y^{(1)}_0=0,\tag{\(0.2\)}\] where \((a_n)\), \((b_n)\), \((p_n)\), \((q_n)\), \(n\in\mathbb{N}\) are complex sequences, \(\gamma_i\), \(\beta_i \in\mathbb{C}\), \(i = 0, 1, 2\) and \(\lambda\) is a eigenparameter. Discussing the eigenvalues and the spectral singularities, we prove that the BVP (0.1), (0.2) has a ifinite number of eigenvalues and spectral singularities with a finite ultiplicities, if \[\sum_{n=1}^\infty \exp(\varepsilon n^\delta)(|1-a_n|+ |1+b_n|+ |p_n|+ |q_n|)<\infty,\] holds, for some \(\varepsilon\) and \(\frac{1}{2}\le\delta\le 1\).