  Recent experiments on spontaneous emission of atomic matter waves open a new
window into the behavior of quantum emitters coupled to a waveguide. Here we
develop an approach based on infinite products to study this system
theoretically, without the need to approximate the band dispersion relation of
the waveguide. We solve the system for a one-dimensional array of one, multiple
and an infinite number of quantum emitters and compare with the experiments.
This leads to a detailed characterization of the decay spectrum, with a family
of in-gap bound states, new mechanisms for enhanced Markovian emission
different from superradiance, and the emergence of matter-wave polaritons.
