Eigenvalues and scattering properties of difference operators with impulsive condition

Abstract

In this work, we are concerned with di¤erence operator of second order with impulsive condition. By the help of a transfer matrix M , we present scattering function of corresponding operator and examine the spectral properties of this impulsive problem.

Principal functions of impulsive difference operators on semi axis

“…To prove the next lemma and theorem, we need the following theorem given in [20]: (ii) If + + + = 0;…”

“…Proof. Under the condition (20), it follows from (8) that jA j;m j ce 4 m ; j = k + 1; k + 2; m = 1; 2; :::;…”

“…where c > 0 is a constant and m 2 denotes the integer part of m 2 : On the other hand, two solutions of impulsive di¤erence boundary value problem (3) (5) are stated in [20] as…”

“…On the other hand, for di¤erence equations, a lot of spectral results have been investigated in the literature [12,13]. Moreover, for the spectral theory of di¤erence equations, [14,15,16,17,18,19,20] are detailed references for the readers. But these references are all related to general boundary conditions.…”

“…But these references are all related to general boundary conditions. Hence, the aim of this paper is to study some spectral properties of the impulsive di¤erence operator mentioned in [20] which is still an uninvestigated problem in literature.…”