  This paper presents a thorough analysis of 1-dimensional Schroedinger
operators whose potential is a linear combination of the Coulomb term 1/r and
the centrifugal term 1/r^2. We allow both coupling constants to be complex.
Using natural boundary conditions at 0, a two parameter holomorphic family of
closed operators is introduced. We call them the Whittaker operators, since in
the mathematical literature their eigenvalue equation is called the Whittaker
equation. Spectral and scattering theory for Whittaker operators is studied.
Whittaker operators appear in quantum mechanics as the radial part of the
Schroedinger operator with a Coulomb potential.
