SPECTRUM AND SYMMETRIES OF THE IMPULSIVE DIFFERENCE EQUATIONS

This paper deals with the spectral analysis and symmetries of the second order difference equations with impulse. We determine a transfer matrix and this allows us to investigate the locations of eigenvalues and spectral singularites of the difference operator generated in `2(Z).


Introduction
As is well known, impulsive equations appear as a natural description of the observed evolution phenomena in several real world problems [1][2][3][4][5]. The books on the subject of impulsive equations by Samoilenko [1] summarize and organize the theories and applications of impulsive equations and have a great contribution to the theory. In recent years, impulsive points have been a subject of both theoretical and experimental research and we observe an increasing interest about this area. Theory of impulsive di¤erential equations has been motivated by a number of applied problems (control theory, population dynamics, chemotherapeutic treatment in medicine and some physics problems). Besides, impulsive di¤erence equations are basic tools to do investigations in numerical analysis having applications in economics, social sciences, biology, engineering, etc... For the mathematical theory of impulsive di¤erence equations, we refer to [6][7][8]. The spectral analysis of self-adjoint (Hermitian) and non-selfadjoint (non-Hermitian) di¤erence operators were investigated in detail by several authors [9][10][11][12]. But the theory of impulsive SPECTRUM AND SYM M ETRIES OF THE IM PULSIVE DIFFERENCE EQ UATIONS 39 di¤erence equations is a new and important branch of operator theory which is interesting and useful.
In spectral theory, the concept of a spectral singularity of a second-order linear di¤erential operator has been known since the pioneering work of Naimark [13] and recently studied by Guseinov [14]. It was proved that the spectral singularities are the spectral points of continuous spectrum and they spoil the completeness of the eigenfunctions of certain non-Hermitian operators. Hermitian operators have no spectral singularities and they have real spectrums. But the reality of the spectrum of an operator doesn't necessarily mean that it is Hermitian. A class of non-Hermitian operators which are called P,T , and PT -symmetric, have real spectrum [15,16]. A kind of such operators was studied in the recent years and the physical meaning and potential applications of spectral singularities were understood quite recently [14,17]. In [17], Mostafazadeh considers the second-order time-independent Schrödinger equation with a general point interaction x 0 introduces the two-component wave function and , + de…ne respectively the restrictions of the solution of (1.1). His study becomes for us a tool to form the impulsive di¤erence equation with general point interaction at a single point n = 0. The aim of present article is to explore the eigenvalues and spectral singularities depending on the choice of the coupling constants a; b; c; d for impulsive di¤erence point interaction.

Spectral Singularities and Eigenvalues of the Impulsive Discrete Operator
We consider the second-order di¤erence equation y n 1 + y n+1 = y n ; n 2 Zn f 1; 0; 1g (2.1) with impulsive condition where a; b; c; d are complex numbers and := z + 1 z is a spectral parameter. Also, 4 denotes the forward di¤erence operator and 5 denotes the backward di¤erence operator, i.e. 4y n := y n+1 y n ; 5y n := y n y n 1 : fy n g and fy n + g be respectively the restrictions of the solution fy n g to the sets of negative and positive integer numbers, i.e. y n (z) := y n (z); n 2 Z ; y + n (z) := y n (z); n 2 Z + : Clearly, n = 0 is the interaction (impulsive) point and matrix B is used to continue the solution from negative integer numbers to the positive integer numbers. Now, let z 2 Cnf 1; 0; 1g and take into account this interaction for (2.1). By the help of linearly independent solutions of (2.1), we can write the general solution as y n (z) = A z n + B z n ; n 2 Z ; y + n (z) = A + z n + B + z n ; n 2 Z + ; where A and B are constant coe¢ cients. If we introduce the two-component wave function Now, consider the left-and right-going scattering solutions of (2.1)-(2.2) that we denote by y l n and y r n , respectively. They are expressed as A z n + B z n ; n ! 1; where A and B are complex coe¢ cients. Let L denote the operator generated by (2.1)-(2.2) in`2(Z). By using the de…nition of the Wronskian of the Jost solutions of the operator L acting in`2(Z), we can give the following theorem [18]: Theorem 2.1. The following asymptotics hold: Proof. The left-and right-going scattering solutions are de…ned in terms of their asymptotic behaviors y l n (z) ! z n ; n ! +1; y r n (z) ! z n ; n ! 1 and so we obtain This implies that Similarly, for the right-going scattering solution, we get Now, we can express the left-and right-going scattering solutions of (2.1)-(2.2) as : Since the Wronskian of linearly independent solutions is independent of n, (2.9) can be used to perform the Wronskian of the Jost solutions for n ! +1 and for n ! 1.
(i) For n ! +1, we get W y l n ; y r n (z) = z n M 12 z n+1 + M 22 z n 1 z n+1 (M 12 z n + M 22 z n ) Therefore, the proof is completed.

SPECTRUM AND SYM M ETRIES OF THE IM PULSIVE DIFFERENCE EQ UATIONS 43
The transfer matrix for a piecewise continuous scattering potentials has unit determinant [17]. For the point interaction (2.2), we consider det M = det B = ad bc = 1: (2.10) The interactions violating this condition are called anomalous point interactions [17]. Using Theorem 2.1 and (2.10), we have the following: Corollary 2.2. The necessary and su¢ cient condition to investigate the eigenvalues and spectral singularities of the operator L is to investigate the zeros of the function M 22 .
Since the spectral singularities and eigenvalues of L correspond to values for which M 22 (z) = 0, we need to examine the zeros of the function M 22 , i.e., and ss (L) will denote the eigenvalues and spectral singularities of L, respectively. Therefore, by the de…nitions of eigenvalues and spectral singularities of an operator, we can write [11,13], where D 1 := fz : 0 < jzj < 1g and D 0 := fz : jzj = 1g.
In order to examine the zeros of (2.11), we consider the following cases. In particular, M is independent of , M 22 vanishes identically, and the interaction is anomalous for a 6 = i.

P, T , and PT -Symmetries
In this section, we examine the consequences of imposing P, T , and PTsymmetries on the point interaction (2.2) and their spectral singularities, eigenvalues.
3.1. P-Symmetry. De…nition 3.1. Let P be the parity (re ‡ection) operator acting in the space of complex sequences y = fy n g ; y n : Z ! C. Then for all n 2 Z, we have where the action of P on a two-component wave function is de…ned componentwise.
Hence, it is easy to verify that P 4 y n = 5 y n P 5 y n = 4 y n : This gives a consequence that 3 = B 3 B: Now, let (3.2) holds. In exactly the same way, we can prove that the interaction (2.2) has P-symmetry.
In terms of the entries of B, we can give that the point interaction (2.2) has P-symmetry as the following theorem: Proof. Assume that (2.2) has P-symmetry, we obtain  and an eigenvalue exists whenever 0 < jc 2j < 2:

T -Symmetry.
De…nition 3.6. Let T be the time-reversal operator acting on the space of complex sequences y = fy n g, y n : Z ! C. Then for all n 2 Z, we have T y n := y n ; where " " denotes the complex conjugate of y.

De…nition 3.7. The point interaction (2.2) is the time-reversal invariant (or has
where the action of T on a two-component wave function is de…ned componentwise. and it is clear from that These relations give us the following cases: Consequently, we also give the both cases with a single expression such that 8 > > < > > : where 1 := ( 1) m ; 2 := 1 ; 0 r b r c 1 1; r b r c > 1 and thus, the transmission matrix B is given as where + = 2 k, k 2 Z. or ja + bj 6 = 1 and ja + 2bj < 1 < jaj : (ii) Suppose that b = 0 and a + d = trB 6 = 0. Then, we have a spectral singularity if ja + c + dj = ja + dj and an eigenvalue if 0 < ja + c + dj < ja + dj :

Conclusions
In this paper, we study the spectral analysis of an impulsive di¤erence operator in`2(Z) generated by a second-order di¤erence equation with an impulsive interior point. In literature, there exist a lot of paper investigating the properties of eigenvalues and spectral singularities of di¤erence operators (see [9][10][11]), but this study di¤ers from the others with some aspects. First of all, none of the di¤erence equations considered in these studies have a discontinuous point. We handle the di¤erence equation with one discontinuity given at n = 0. Furthermore, unlike the well known methods, we determine a transfer matrix which enables us to …nd the locations of eigenvalues and spectral singularities by the help of the zeros of a quadratic polynomial. The rest of paper is devoted to a detailed analysis of certain symmetries that the impulsive condition possesses. This paper is the …rst one that points up the e¤ects of a single interaction point of an impulsive di¤erence boundary value problem. In further research, one can study the general form of Equation (2.1) with more interaction points.