The one-dimensional Schrödinger–Coulomb operator with definite parity

Abstract

As a model for one-dimensional hydrogenic systems, the Schrödinger operator with the singular potential −|x|−1 is studied by analyzing it in its reducing subspaces corresponding to even and odd parity. We derive spectral properties common to all self-adjoint extensions of the reduced operators and extract the boundary conditions at x = 0 specific to the different self-adjoint extensions. A transcendental equation for the eigenvalues is derived which leads to a construct that we call the ‘spectral helix’. For the distinguished self-adjoint extension with Dirichlet boundary conditions, virial and supersymmetry relations are established.