  In this paper we analyze the behavior of quantum random walks. In particular
we present several new results for the absorption probabilities in systems with
both one and two absorbing walls for the one-dimensional case. We compute these
probabilites both by employing generating functions and by use of an
eigenfunction approach. The generating function method is used to determine
some simple properties of the walks we consider, but appears to have
limitations. The eigenfunction approach works by relating the problem of
absorption to a unitary problem that has identical dynamics inside a certain
domain, and can be used to compute several additional interesting properties,
such as the time dependence of absorption. The eigenfunction method has the
distinct advantage that it can be extended to arbitrary dimensionality. We
outline the solution of the absorption probability problem of a
(d-1)-dimensional wall in a d-dimensional space.
