We consider one-dimensional quantum many-body systems with pair interactions in external fields and present conditions under which ground state wave functions of product type can be found. An overview of a list of such systems is given. This list contains a system whose ground state has a short-range behavior at the two-particle coalescence points which looks similar to the behavior of three-dimensional many-electron systems with Coulomb interactions. This property of the ground state makes this system interesting as a model problem for the test or comparison of approximation schemes for many-electron wave functions. We will focus on this particular many-body system and discuss its properties in detail.