We provide a theoretical framework to describe the interaction of a propagating guided matter wave with a localized potential in terms of a quantum scattering approach in a confined environment. This interaction generates entangled states for which the longitudinal and transverse degrees of freedom are correlated. The number of terms of the entangled state is dictated by the incident energy. We analyze this scattering analytically under the Born approximation using a Gaussian localized potential. In this limit, it is possible to engineer the potential and achieve the 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 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 are, finally, discussed.