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

On a finite momentum grid with N   integration points pnpn and weights wnwn (n=1,…,Nn=1,…,N) the Similarity Renormalization Group (SRG) with a given generator G unitarily evolves an initial interaction with a cutoff λ   on energy differences, steadily driving the starting Hamiltonian in momentum space Hn,m0=pn2δn,m+Vn,m to a diagonal form in the infrared limit (λ→0λ→0), Hn,mG,λ→0=Eπ(n)δn,m, where π(n)π(n) is a permutation of the eigenvalues EnEn which depends on G  . Levinson's theorem establishes a relation between phase-shifts δ(pn)δ(pn) and the number of bound-states, nBnB, and reads δ(p1)−δ(pN)=nBπδ(p1)−δ(pN)=nBπ. 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 S01 and S13 channels.