  Discrete-time quantum walks are well-known for exhibiting localization, a
quantum phenomenon where the walker remains at its initial location with high
probability. In companion with a joint Letter, we introduce oscillatory
localization, where the walker alternates between two states. The walk is given
by the flip-flop shift, which is easily defined on non-lattice graphs, and the
Grover coin. Extremely simple examples of the localization exist, such as a
walker jumping back and forth between two vertices of the complete graph. We
show that only two kinds of states, called flip states and uniform states,
exhibit exact oscillatory localization. So the projection of an arbitrary state
onto these gives a lower bound on the extent of oscillatory localization. Thus
we completely characterize the states that oscillate under the quantum walk.
