| This thesis work has investigated the Renormalization Group theory for the nucleon-nucleon interaction. Conventional nuclear many-body calculations have the following sources of non-perturbative physics: (1) a strongly repulsive short-range interaction, (2) a tensor force, e.g. from pion exchange, which is highly singular at short-distances, (3) the presence of low-energy bound states or nearly bound states (in the S waves).; The RG approach exploits the insensitivity of low-energy processes to the details of the high-energy physics. Using any of the high-precision potentials as input, the high-momentum intermediate states in the Lippmann-Schwinger equation for the T matrix in a particular partial-wave are cut-off at Λ. The details of the physics beyond this limit of resolution are integrated out and included in the potential by requiring that the half-off shell T matrix elements be independent of the cut-off Λ. This requirement leads to a low-momentum potential "V low k", which is energy independent.; The choice of the regulator which cuts off the high momentum intermediate states is investigated. Sharp cut-offs, though straight forward, lead to convergence issues in few-body calculations that are eliminated using smooth regulators. The construction of low-momentum potentials using a smooth regulator is explored in detail. In the course of this study, a three-step process to calculate Vlow k requiring the cut-off independence of the fully-off shell T matrix elements has been established and this yields better numerical stability than the energy-independent RG.; The complex eigenvalues (Weinberg eigenvalues) of the operator G0(z)V, which appears in the Lippmann-Schwinger equation, are a useful tool for investigating the convergence of the Born series. Weinberg eigenvalues for V low k potentials, including chiral effective theory potentials, have been investigated as a function of cut-off. The studies reveal the density and/or scale dependence of the sources of non-perturbative physics. The in-medium eigenvalues near the Fermi surface give a good estimate of the pairing gaps. Using two-particle Nambu-Gorkov propagators, the eigenvalue equation at E = 2epsilonF is the gap equation and the eigenvectors corresponding to the largest eigenvalue gives the first approximation to the gap function Delta(k), which can be further iterated using the BCS gap equation to give self-consistent gaps. |
