Investigation of spectrum and scattering function of impulsive matrix difference operators

Abstract

In this paper, we consider a second-order impulsive matrix difference operators. Using the asymptotic and analytical properties of the Jost function, we investigate eigenvalues, spectral singularities, resolvent operator, spectrum and scattering function of this problem. Finally, we study spectrum and scattering function of an unperturbated impulsive matrix difference equation.

Levinson theorem for discrete Schrödinger operators on the line with matrix potentials having a first moment

“…Next recall Definition 3 introducing the scattering matrix S z for z ∈ D by (7), provided that M z ± are invertible. This is the case on the unit circle:…”

“…Scattering in a periodic background is treated in [14]. Works on the scattering theory for the matrix-valued case are scarce [23,6,7], but the latter two also construct Jost solutions and a scattering matrix under a moment condition similar as is done below. What is missing in [7], however, is the fine analysis of the analytic behavior of the Jost solutions and the scattering matrix at the band edges so that the authors could not conclude that there is a finite number of bound states nor analyze half-bound states nor prove a Levinson theorem.…”

“…Works on the scattering theory for the matrix-valued case are scarce [23,6,7], but the latter two also construct Jost solutions and a scattering matrix under a moment condition similar as is done below. What is missing in [7], however, is the fine analysis of the analytic behavior of the Jost solutions and the scattering matrix at the band edges so that the authors could not conclude that there is a finite number of bound states nor analyze half-bound states nor prove a Levinson theorem. For scattering theory for continuous one-dimensional Schrödinger operators with a matrix-valued potential, there is also abundant literature, most of which is cited in the recent monograph by Aktosun and Weder [4].…”

Scattering analysis of a quantum impulsive boundary value problem with spectral parameter

“…Impulsive conditions are also called transmission conditions, jump conditions, interface conditions and point conditions in literature [14,21,26,30,31,34,35]. Recently, some researchers have paid more attention to scattering problems of differential and difference equations with impulsive conditions [4,[10][11][12][13]20]. Differently from these papers, we will consider a quantum impulsive boundary value problem (QIBVP) in this paper and differently from [4], this QIBVP consists spectral parameter in boundary condition.…”

Spectral singularities of an impulsive Sturm-Liouville operators

“…Recently, many authors have examined impulsive differential equations in detail, because impulsive differential equations have been used in many scientific phenomena such as heart beat, population dynamics, atomic physics, mathematical economics, ecology, engineering, medicine and so forth [13,15,19]. Bairamov et al, Yardimci and Erdal investigated scattering analysis and spectral theory of different kinds of impulsive Sturm-Liouville equations [2,5,6,8,11,24]. Different from these studies, in this paper, we consider the Sturm-Liouville equation with complex valued potential and impulsive condition in matrix form.…”