  We consider the solution of the quantum Coulomb problem in one dimension with
the most general connection condition at the origin. The divergence of the
derivative of the wave function at the origin invalidates the standard current
conservation approach. We explore two approaches, Wronskian self-adjoint
extension method and cutoff regularization method, and establish their mutual
relations, thereby clarifying the physical contents of the connection
parameters. We show how to realize exotic non-Loudon connection conditions,
entailing the realization of non-Rydberg spectrum.
