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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

(1)(d 2 /dt 2 +K 2 )u(r)=v(r)u(r)- 0 χ(r,r ' )v(r ' )u(r ' )dr ' ,whereχ(r,r ' )=(1/π)[sin(r-r ' )/(r-r ' )-sin(r+r ' )/(r+r ' )]·

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:
45J05Integro-ordinary differential equations
81Q40Bethe-Salpeter and other integral equations in quantum theory
81V05Strong interaction, including quantum chromodynamics
45B05Fredholm integral equations
45C05Eigenvalue problems (integral equations)