The infrared limit of the SRG evolution and Levinson's theorem

On a finite momentum grid with N integration points p(n) and weights w(n) (n = 1,..., N) the Similarity Renormalization Group (SRG) with a given generator G unitarily evolves an initial interaction with a cutoff lambda on energy differences, steadily driving the starting Hamiltonian in momentum space H-n,m(0) = p(n)(2)delta(n,m) + V-n,(m) to a diagonal form in the infrared limit (lambda -> 0), H-n,m(G,lambda -> 0) = E-pi(n) delta(n),(m), where pi(n) is a permutation of the eigenvalues E-n which depends on G. Levinson's theorem establishes a relation between phase-shifts delta(p(n)) and the number of bound-states, n(B), and reads delta(p(1)) - delta(p(N)) = n(B)pi. We show that unitarily equivalent Hamiltonians on the grid generate reaction matrices which are compatible with Levinson's theorem but are phase-inequivalent along the SRG trajectory. An isospectral definition of the phase-shift in terms of an energy-shift is possible but requires in addition a proper ordering of states on a momentum grid such as to fulfill Levinson's theorem. We show how the SRG with different generators G induces different isospectral flows in the presence of bound-states, leading to distinct orderings in the infrared limit. While the Wilson generator induces an ascending ordering incompatible with Levinson's theorem, the Wegner generator provides a much better ordering, although not the optimal one. We illustrate the discussion with the nucleon-nucleon (NN) interaction in the S-1(0) and S-3(1) channels. (C) 2014 The Authors. Published by Elsevier B. V.