Abstract
The currently accepted diagram-cancellation prescription for the single-particle potential in the reaction-matrix theory of finite nuclei is studied. It is shown that, under very general and weak assumptions, the potential defined by this prescription is a Hilbert-Schmidt operator of finite rank (a “separable” potential). Its individual “form factors” are closely related to the wave functions of the occupied states. All unoccupied states belong to the continuous spectrum. The structure of these continuum states is exhibited in an exact closed form which is determined uniquely by the A occupied states and represents a solution to the “orthogonalized plane wave” problem of Brueckner-Hartree-Fock theory. Using the harmonic-oscillator approximation for occupied-state wave functions (but not energies), closed analytic expressions are derived for the continuum wave functions, and quantitative results are presented on their deviations from plane waves as a function of energy. The nuclear-matter approximation for the Pauli projection operator is derived without reference to nuclear-matter concepts, and numerical values of the effective Fermi momentum are presented. The finite-rank Hilbert-Schmidt property is shown to persist in a good approximation even for certain extensions of the potential definition considered here. We discuss the advantages of using finite-rank models for the singleparticle potential in nuclear-reaction theories.
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