The one-dimensional hydrogen atom revisited

The one-dimensional Schrodinger hydrogen atom is an interesting mathematical and physical problem for the study of bound states, eigenfunctions, and quantum-degeneracy issues. This one-dimensional physical system has given rise to some intriguing controversy for 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 Schrodinger equation at the origin.