  This paper presents the spectral analysis of 1-dimensional Schroedinger
operator on the half-line whose potential is a linear combination of the
Coulomb term 1/r and the centrifugal term 1/r^2. The coupling constants are
allowed to be complex, and all possible boundary conditions at 0 are
considered. The resulting closed operators are organized in three holomorphic
families. These operators are closely related to the Whittaker equation.
Solutions of this equation are thoroughly studied in a large appendix to this
paper. Various special cases of this equation are analyzed, namely the
degenerate, the Laguerre and the doubly degenerate cases. A new solution to the
Whittaker equation in the doubly degenerate case is also introduced.
