On Radial Schrödinger Operators with a Coulomb Potential

This paper presents a thorough analysis of one-dimensional Schrödinger operators whose potential is a linear combination of the Coulomb term 1 / r and the centrifugal term \$\$1/r^2\$\$ 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 on \$\$L^2(\backslashmathbb \R\_+)\$\$ L 2 ( R + ) 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 Schrödinger operator with a Coulomb potential.