An integral transform solution for the one-dimensional hydrogen atom

The one-dimensional Schroedinger equation for the hydrogen atom is shown to be solvable in terms of an integral transform originally due to Bateman. Two sets of wave functions are obtained; the first set consists of polynomials that correspond to the integer principal quantum numbers and the second set consists of infinite series that corresponds to the non-integer principal quantum numbers. The results prove conclusively that all states of the one-dimensional hydrogen atom are non-degenerate and that the concept of degeneracy due to different parity about the origin does not apply here. (author)