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A closed form solution of the s-wave Bethe-Goldstone equation with an infinite repulsive core. (English) Zbl 0595.45014

The author discusses the s-wave solution of the Bethe-Goldstone equation for the interaction of two nucleons characterized by a potential with an infinite repulsive core. By introducing the dimensionless variables, the considered equation reduces to the form

$\left(1\right)\phantom{\rule{1.em}{0ex}}\left({d}^{2}/d{t}^{2}+{K}^{2}\right)u\left(r\right)=v\left(r\right)u\left(r\right)-{\int }_{0}^{\infty }\chi \left(r,{r}^{\text{'}}\right)v\left({r}^{\text{'}}\right)u\left({r}^{\text{'}}\right)d{r}^{\text{'}},\phantom{\rule{4.pt}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}\chi \left(r,{r}^{\text{'}}\right)=\left(1/\pi \right)\left[sin\left(r-{r}^{\text{'}}\right)/\left(r-{r}^{\text{'}}\right)-sin\left(r+{r}^{\text{'}}\right)/\left(r+{r}^{\text{'}}\right)\right]·$

Above equation is transformed to a Fredholm integral equation of the second kind and by an application of Hilbert-Schmidt theorem the author obtains a closed form of the solution of (1) in terms of angular prolated spheroidal wave functions. The author also indicates that the asymptotic result for the case of small core radius is in excellent agreement with the known results obtained via an approximate iterative procedure.

Reviewer: Enhao Yang

##### MSC:
 45J05 Integro-ordinary differential equations 81Q40 Bethe-Salpeter and other integral equations in quantum theory 81V05 Strong interaction, including quantum chromodynamics 45B05 Fredholm integral equations 45C05 Eigenvalue problems (integral equations)