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The infrared limit of the Similarity Renormalization Group evolution and Levinson's theorem

On a finite momentum grid with N integration points and weights the Similarity Renormalization Group (SRG) with a given generator G unitarily evolves an initial interaction with a cutoff on energy differences. This steadily drives the starting Hamiltonian in momentum space to a diagonal form in the infrared limit corresponding to a permutation of the eigenvalues and depends on G. Levinson's theorem establishes a relation between phase-shifts and the number of bound-states. 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 in order 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 1S0 and 3S1 channels.