The hyperspherical-expansion approach and shell models

By means of Talmi integrals the matrix elements of nuclear interaction in the set of radial differential equations of the hyperspherical-expansion approach are related to the elements of usual shell models. The introduced average field equated to the diagonal element of the core reproduces the form of the effective (two-body) cut-off oscillator. The corresponding eigenvalue problem with a flat (heavy nuclei) and rounded (light nuclei) alternative for the well is solved and compared with the phenomenological relations: the asymptotic saturation can be achieved for the flat well. Using these results the starting set is transformed into the set of integral equations by means of the Green's functions. Then it is, successively, reduced to the eigenvalue equations of usual shell models identical for both the well alternatives. Compared with the usual models some improvements apply and the oscillator length is given automatically.