Spectral properties of impulsive quantum difference equations

The first chapter is devoted to the introduction. In the second chapter, some well known concepts of functional analysis and spectral theory are mentioned and an example of an eigenvalue problem is presented. In the third chapter, some generalizations and consequences are given for dynamic equations on time scales containing $ q$--difference equations as a special case. Later, the $ q $--analogous of the Sturm--Liouville equation is obtained and some spectral properties of the second order $ q $--difference operator corresponding to that second--order equation are mentioned. The fourth chapter consists of two sections. In the first section, the impulsive $ q$--difference operator is defined, theory on the eigenvalues and the spectral singularities of this operator is given. In the second section, some special cases are examined, where the impulsive condition at the point $ t=1 $ has certain symmetries. The fifth and last chapter is devoted to the conclusion and recommendations.