ORTHOGONALITY CRITERIA FOR SINGULAR STATES AND THE NONEXISTENCE OF STATIONARY STATES WITH EVEN-PARITY FOR THE ONE-DIMENSIONAL HYDROGEN-ATOM

With the aid of two linearly independent Whittaker functions, Loudon o btained the solutions with even and odd parity for the one-dimensional hydrogen atom. Applying the Schwarz inequality, Andrews made an objec tion to Loudon's ''ground state.'' Either solving the problem in the m omentum representation or basing our work on the theory of singular in tegral equations, we have proved that these solutions with even parity do not exist. Due to its importance related to the nondegeneracy theo rem and to the study of the exciton and Wigner crystal (by electron ga s above the helium surface), we have reexamined this problem in the co ordinate representation by means of the orthogonality criterion for si ngular states and the natural connection condition of the wave functio n'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 impor tance of the orthogonality criterion but also generalized its applicat ion, including the singular states with poles, essential singular poin ts, phase angle uncertainty, and the logarithmic singularity of deriva tives.