Green's functions of the Schrödinger equation for the simplest systems

Closed analytic representations of the Green's functions of the Schrödinger equation are considered for an harmonic oscillator (linear and three-dimensional isotropie oscillator), the Morse oscillator, the generalized Kepler problem (the Kratzer potential), and for the double symmetric potential well $V(x)=\frac{m\omega^2}{2}(|x|-R)^2$. The coordinate representation of the Green's function is expressed in a form convenient for applications. These models, like those of free motion and the hydrogen atom (for which closed expressions for the Green's functions are known), belong to the class of problems for which the Schrödinger equation can be reduced to the canonical form of the confluent hypergeometric equation.