  When classically searching a database, having additional correct answers
makes the search easier. For a discrete-time quantum walk searching a graph for
a marked vertex, however, additional marked vertices can make the search harder
by causing the system to approximately begin in a stationary state, so the
system fails to evolve. In this paper, we completely characterize the
stationary states, or 1-eigenvectors, of the quantum walk search operator for
general graphs and configurations of marked vertices by decomposing their
amplitudes into uniform and flip states. This infinitely expands the number of
known stationary states and gives an optimization procedure to find the
stationary state closest to the initial uniform state of the walk. We further
prove theorems on the existence of stationary states, with them conditionally
existing if the marked vertices form a bipartite connected component and always
existing if non-bipartite. These results utilize the standard oracle in
Grover's algorithm, but we show that a different type of oracle prevents
stationary states from interfering with the search algorithm.
