THE CUBIC EIGENPARAMETER DEPENDENT DISCRETE DIRAC EQUATIONS WITH PRINCIPAL FUNCTIONS

Abstract

Let us consider the Boundary Value Problem (BVP) for the discrete Dirac Equations ( an+1y (2) n+1 + bny (2) n + pny (1) n = y(1) n an1y (1) n1 + bny (1) n + qny (2) n = y(2) n ; n 2 N; (0.1) (0 + 1 + 2 2 + 3 3 )y (2) 1 + (0 + 1 + 2 2 + 3 3 )y (1) 0 = 0; (0.2) where (an); (bn); (pn) and (qn); n 2 N are complex sequences, i ; i 2 C; i = 0; 1; 2 and is a eigenparameter. Discussing the eigenvalues and the spectral singularities, we prove that the BVP (0.1), (0.2) has a Önite number of eigenvalues and spectral singularities with a Önite multiplicities, if X1 n=1 exp("n ) (j1 anj + j1 + bnj + jpnj + jqnj) < 1; holds, for some " > 0 and 1 2.