The periodic unitary transition operators are certain unitary operators defined over integer lattices in Euclidean spaces, which have been introduced as a generalization of the so-called "quantum walks with constant coins”.

Quantum walks have been discovered in the area of quantum physics and computer sciences around 2000 and studied intensively in probability theory and discrete geometric analysis as well as the area mentioned above. Quantum walks are non-commutative analogue of classical random walks. But, their behavior is quite different. Even one-dimensional quantum walks defined as an analogue of the so-called lazy random walk have an interesting phenomenon called“localization”.

In this talk, the localization for periodic unitary transition operators will be discussed from a view points of spectral theory. Also a criterion for the localization in terms of given “coin matrices” for certain class of operators, which is obtained by a joint work with T. Komatsu, will be given.