Model of an unstable system with Coulomb interaction distorted at small distances

An exactly solvable model of an unstable system with Coulomb plus nuclear interaction is proposed. The system under consideration consists of two pairs of particles which can transit into each other. The heavy particles interact via the Coulomb plus short-range separable potential. The nondiagonal potential which is responsible for the transition of a pair of heavy particles to a pair of light ones is also taken in a separable form. The particles in the decay channel are supposed to be free. Closed expressions are obtained for the T-matrix, nuclear and Coulomb-modified scattering lengths. A detailed analysis of the equation for the shift of the atomic S levels of heavy particles is presented