# What is the BG Equation?

About 45 years ago or so I was a Physics department staff computer programmer when I was a physics major undergrad. I worked with a professor doing research on nuclear shell models -- he was attempting to develop a numerical method that would solve particular potentials using the BG equation as the mathematical foundation. My job was to write the programs, run jobs, plot graphic output on a nice 27 inch Calcomp flat bed plotter, and other such things. However, whatever I might have known, I have forgotten.

But, I do have one form of this equation written below: $$\left[\lambda + \frac{\mathrm{d}^2}{\mathrm{d}\rho^2}-V(\rho)\right]U(\rho) = (\lambda-\lambda^0)R(\rho) \langle R {\mid} U \rangle - \sum_a C^aR^a(\rho) \langle R^a {\mid} \nu {\mid} U \rangle$$ Unfortunately, I do not have definitions of the variables other than I knew that $V(\rho)$ is of course the potential and I believe the $\lambda$ are eigenvalues.

The professor also notes in a hand-written note he prepared for me that the numerical method we were investigating is based on the Fox-Goodwin method. He wrote in the margin, just barely legible that the Fox-Goodwin method he refers to is described in a paper cited as "Proc. of Cambridge Phil. Soc. 45 (1949) 373".

So, with this skimpy information out of the past, can someone give me more information on this so-called BG equation? I believe that B and G are the initials of last names (?) of the authors of some earlier paper.

By the way, I have googled various aspects of these facts I know and I found some possible hits but they referred to papers that I could not find on-line and probably predated any form of publication on the Internet.

• According to Google, BG refers to Bethe-Goldstone. May 28, 2016 at 2:56
• Yes, I found a link for Bethe-Goldstone before but it was a reference to a paper I had no access to. However, just googling it again, I did find a book reference that seems to give me a lot of information that I didn't have before: books.google.com/… May 28, 2016 at 3:00
• The BG equation is frequently called the BBG (Brueckner-Bethe-Goldstone) equation. It is used to calculate corrections to the mean field or Hartree-Fock model of a many body system. May 28, 2016 at 20:47