Smooth amplitude-phase formulation of the Schrödinger equation based on the Ermakov invariant

The amplitude-phase formulation of the one-dimensional Schrödinger equation is investigated within the context of Ermakov systems. The boundary conditions for amplitude functions corresponding to bound states are given in terms of the Ermakov invariant and a related constant, which also monitors the behavior of the accumulated phase function. A procedure leading to the numerical construction of smooth, nonoscillating amplitude and phase functions is proposed, and illustrated in the case of the harmonic oscillator and the centrifugal Coulomb potential. The use of this procedure as a tool to define radial basis functions for bound channels within the framework of scattering theory is discussed.