As has been presented in a previous paper, the use of hyperspherical coordinates in research on a system of electrically charged particles carries mathematical complications into the evaluation of certain kinds of hyperangular interaction integrals. These integrals contain the hyperangular interaction potential and various powers of the inverse of the total angular momentum operator on the space of hyperharmonics, from which the zeroth order hyperharmonic is excluded. In this work the simplest one of these integrals has been taken into consideration. After some intermediate steps, it has been shown that it can be expressed in terms of elementary functions of the cosine of the angle (γ) between the hyperaxes of the potential term for an odd number of particles. In the case of an even number of particles, these integrals can be given in terms of a generalized hypergeometric function of the same argument (γ) and its derivatives.
In recent years it has been shown that the use of hyper spherical coordinate representation of the Schrodinger equation for electrically charged particles necessitates the evaluation of certain kinds of hyper angular interaction integrals. The analytic evaluation of rather simple cases has also been accomplished. On the other hand certain numerical devices have been utilized and the complications that have arisen have been discussed for their computation. We have attempted in this work to find upper and lower bounds for all types of zeroth order hyperangular interaction integrals having Coulombic potentials. A nesting procedure has been developed for obtaining close bounds which can possibly be used to evaluate the desired value of integrals under consideration. In this context a theorem has been established for the evaluation of a similar type of integral and possible ways towards the generalization of the theorem have also been discussed. For three particle systems some applications have also been presented.
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