Exactly solvable potentials with Dirac delta point interactions

Abstract

A general method to obtain s-wave bound states of the Schrödinger equation for analytically solvable potentials together with a finite number of the Dirac delta functions is presented for three-dimensional systems. The eigenvalue equations for the finite number of the Dirac delta decorated Woods-Saxon potential and Morse potential are obtained and the eigenvalue equations are calculated numerically for one Dirac delta function. The eigenvalue equation is solved numerically for two delta function case for the Woods-Saxon potential. The change of the ground state energies as a function of strength and locations of the Dirac delta functions are investigated for these potentials. For a hydrogen molecule, which is described by the Morse potential, transition probabilities between the initial state of the Hamiltonian and the final state of the perturbed Hamiltonian for a sudden, very local perturbation are calculated. The conditions for the number of bound states for given parameters of the Woods-Saxon and the Morse potentials are analyzed. Our results for the Woods-Saxon potential can be used to model very short range interactions for atomic and nuclear systems. Our method is applied to hydrogen molecule, which is described by the Morse potential, in C60 fullerene cage and the stabilization energy is calculated by representing the interaction potential between H2 and C60 by the Dirac delta interaction. Interatomic interactions that are described by the Morse potential together with local deformations can be investigated by using our model.