One dimensional hydrogen problem with general connection condition at the origin and non-Rydberg spectra

Abstract

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. • The quantum 1D Coulomb problem is formulated on physical and mathematical grounds. • Non-Rydberg spectra are realizable for certain types of 1D hydrogen atoms . • Rectangular regularizations entail different even spectra depending on core behavior.