A closed form solution of the s-wave Bethe-Goldstone equation with an infinite repulsive core.

*(English)*Zbl 0595.45014The 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

$$\left(1\right)\phantom{\rule{1.em}{0ex}}({d}^{2}/d{t}^{2}+{K}^{2})u\left(r\right)=v\left(r\right)u\left(r\right)-{\int}_{0}^{\infty}\chi (r,{r}^{\text{'}})v\left({r}^{\text{'}}\right)u\left({r}^{\text{'}}\right)d{r}^{\text{'}},\phantom{\rule{4.pt}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}\chi (r,{r}^{\text{'}})=(1/\pi )[sin(r-{r}^{\text{'}})/(r-{r}^{\text{'}})-sin(r+{r}^{\text{'}})/(r+{r}^{\text{'}})]\xb7$$

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) |