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Scattering theory is a framework of considerable experimental and theoretical interest that describes the interaction/scattering of waves and particles. There is a wide range of scattering processes that are studied under different branches of physics, such as the collision of particles in classical physics, scattering of electromagnetic waves through a medium in electromagnetism, scattering of low energy quantum particles in quantum mechanics, or creation/annihilation of particles in high energy particle physics.

In this talk, I will discuss the scattering processes modeled by linear and nonlinear time independent Schrödinger equations. First, I will review the concepts such as Jost solutions, transfer matrices, unidirectional reflection and invisibility. In particular, I will represent some theorems regarding the properties of transfer matrices for the linear case and argue the validity of these theorems in the presence of nonlinearity in Schrödinger equations. Also, I will examine the conditions for the existence of blow up solutions for such systems.