Limit theorems and localization of three-state quantum walks on a line defined by generalized Grover coins

In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on a one-dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.