Limit Theorems for the Discrete-Time Quantum Walk on a Graph with Joined Half Lines

We consider a discrete-time quantum walk W_t,κ at time t on a graph with joined half lines mathbb{J}_κ, which is composed of κ half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_t,κ can be reduced to the walk on a half line even if the initial state is asymmetric. For W_t,κ, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_t,κ which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_t,κ. On each half line, W_t,κ converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.