Recently, it has been possible to create metallic chains down to seven-atom long. This has two implications. Experimentally it is necessary to have algorithms that allow converting measured data into microscopic information as, for example, processing a current-voltage curve to obtain electronic density of states (DOS). From a theoretical standpoint, we have a new framework to test extant theories and to further develop them. With this in mind, we present a new theoretical solution to the problem of mapping scanning tunneling microscopy current-voltage curves into DOS-energy curves. Our model is based on a self-consistent solution to the quantum problem of electrons in the presence of an array of attractive centers. The problem, being simpler than its three-dimensional counterparts is solved exactly for DOS and for the current-voltage curves. The main results are that the current-voltage peaks at the position of the energies of the electron in the chain and, when an impurity is present in the chain, its chemical information can be extracted from the value of analytical curves.