Exact Solution of the Schr¨odinger Equation for Composition of Coulomb and Oscillator Potentials

Abstract

The spherically symmetric potential is considered whose dependence on the distance r is described by the smooth composition of Coulomb at r < r0 and oscillator at r > r0 potentials. The boundary distance r0 is determined by the parameters of these potentials. The exact continuous solution of the radial Schr¨odinger equation is expressed in terms of the confluent hypergeometric functions. The discrete energy levels are obtained. The graphic illustrations for the energy spectrum and the radial wave functions are presented.