Discrete impulsive Sturm-Liouville equation with hyperbolic eigenparameter

Let L denote the selfadjoint difference operator of second order with boundary and impulsive conditions generated in ₂( %TCIMACRO2115 %BeginExpansion N %EndExpansion ) by {equation*} aₙ₋₁yₙ₋₁+bₙyₙ+aₙyₙ₊₁=( 2 z) yₙ ,% n∈ %TCIMACRO{{2115} } %BeginExpansion N %EndExpansion \{ k-1,k,k+1\} , {equation*}% {equation*} {array}{c} y₀=0 , \\ \{ {array}{c} yₖ₊₁=θ ₁yₖ₋₁ \\ yₖ₊₁=θ ₂ yₖ₋₁ {array}% . , θ ₁,θ ₂∈ %TCIMACRO{{211d}}% %BeginExpansion R, %EndExpansion {array}% {equation*} where \ aₙ\ _n∈ %TCIMACRO2115 %BeginExpansion N %EndExpansion , \ bₙ\ _n∈ %TCIMACRO2115 %BeginExpansion N %EndExpansion are real sequences and , are respectively forward and backward operators. In this paper, the spectral properties of L such as the resolvent operator, the spectrum, the eigenvalues, the scattering function and their properties are investigated. Moreover, an example about the scattering function and the existence of eigenvalues is given in the special cases, if {equation*} ∑ₙ₌₁∞n( 1-aₙ + bₙ )