Abstract
The Ursell-Mayer cluster expansion method is extended to the case of a q-number model operator for either boson or fermion systems. The many-body problem for a system with strong, hard sphere interactions can be approximately reduced to that for a weakly interacting system without hard sphere interactions. The effective Hamiltonian for the weakly interacting system is given in the form of an expansion series quite analogous to the ordinary cluster expansion. (We call it the “pseudo-cluster expansion” in this paper.) The validity of our expansion method is discussed. It can be proved that in the simplest case we are led to the approximate Bethe-Goldstone equation (or Brueckner approximation). We derive formulas expressing the total energy and the diagonal and non-diagonal matrix elements of operators by the asymptotic wave functions. Possiblities of application of our method are discussed.
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