  We provide a theoretical framework to describe the interaction of a
propagating guided matter wave with a localized potential in terms of quantum
scattering in a confined environment. We analyze how this scattering correlates
the longitudinal and transverse degrees of freedom and work out analytically
the output state under the Born approximation using a Gaussian localized
potential. In this limit, it is possible to engineer the potential and achieve
coherent control of the output channels. The robustness of this approximation
is studied by comparing the stationary scattering theory to numerical
simulations involving incident wave packets. It remains valid in a domain of
weak localized potential that is achievable experimentally. We infer a possible
method to determine the longitudinal coherence length of a guided atom laser.
Then, we detail the non-perturbative regime of the interaction of the guided
matter wave with the localized potential using a coupled channel approach. This
approach is worked out explicitly with a square potential. It yields new
non-perturbative effects such as the occurrence of confinement-induced
resonances. The perspectives opened by this work for experiments are discussed.
