Fixed points of the SRG evolution and the on-shell limit of the nuclear force

We study the infrared limit of the similarity renormalization group (SRG) using a simple toy model for the nuclear force aiming to investigate the fixed points of the SRG evolution with both the Wilson and the Wegner generators. We show how a fully diagonal interaction at the similarity cutoff lambda -> 0 may be obtained from the eigenvalues of the Hamiltonian and quantify the diagonalness by means of operator norms. While the fixed points for both generators are equivalent when no bound-states are allowed by the interaction, the differences arising from the presence of the Deuteron bound-state can be disentangled very clearly by analyzing the evolved interactions in the infrared limit lambda -> 0 on a finite momentum grid. Another issue we investigate is the location on the diagonal of the Hamiltonian in momentum-space where the SRG evolution places the Deuteron bound-state eigenvalue once it reaches the fixed point. This finite momentum grid setup provides an alternative derivation of the celebrated trace identities, as a by product. The different effects due to either the Wilson or the Wegner generators on the binding energies of A = 2, 3, 4 systems are investigated and related to the occurrence of a Tjon-line which emerges as the minimum of an avoided crossing between E-alpha = 4E(t) - 3E(d) and E-alpha = 2E(t). All infrared features of the flow equations are illustrated using the toy model for the two-nucleon S-waves. (C) 2016 Elsevier Inc. All rights reserved.