  The one dimensional Schroedinger hydrogen atom is an interesting mathematical
and physical problem to study bound states, eigenfunctions and quantum
degeneracy issues. This 1D physical system gave rise to some intriguing
controversy over more than four decades. Presently, still no definite consensus
seems to have been reached. We reanalyzed this apparently controversial
problem, approaching it from a Fourier transform representation method combined
with some fundamental (basic) ideas found in self-adjoint extensions of
symmetric operators. In disagreement with some previous claims, we found that
the complete Balmer energy spectrum is obtained together with an odd parity set
of eigenfunctions. Closed form solutions in both coordinate and momentum spaces
were obtained. No twofold degeneracy was observed as predicted by the
degeneracy theorem in one dimension, though it does not necessarily have to
hold for potentials with singularities. No ground state with infinite energy
exists since the corresponding eigenfunction does not satisfy the Schroedinger
equation at the origin.
