Orthogonality criteria for singular states and the nonexistence of stationary states with even parity for the one-dimensional hydrogen atom

Abstract

With the aid of two linearly independent Whittaker functions, Loudon obtained the solutions with even and odd parity for the one-dimensional hydrogen atom. Applying the Schwarz inequality, Andrews made an objection to Loudon's ``ground state.'' Either solving the problem in the momentum representation or basing our work on the theory of singular integral equations, we have proved that these solutions with even parity do not exist. Due to its importance related to the nondegeneracy theorem and to the study of the exciton and Wigner crystal (by electron gas above the helium surface), we have reexamined this problem in the coordinate representation by means of the orthogonality criterion for singular states and the natural connection condition of the wave function's derivatives. We have proved again that all these eigenstates with even parity do not exist. This result is consistent with that of exact solutions in the momentum representation and in the integral equation method canceling divergence. This study not only emphasized the importance of the orthogonality criterion but also generalized its application, including the singular states with poles, essential singular points, phase angle uncertainty, and the logarithmic singularity of derivatives.