Renormalization group and scattering-equivalent Hamiltonians on a coarse momentum grid

We consider the $\pi\pi$-scattering problem in the context of the Kadyshevsky equation. In this scheme, we introduce a momentum grid and provide an isospectral definition of the phase-shift based on the spectral shift of a Chebyshev angle. We address the problem of the unnatural high momentum tails present in the fitted interactions which reaches energies far beyond the maximal center-of-mass energy of $\sqrt{s}=1.4$ GeV. It turns out that these tails can be integrated out by using a block-diagonal generator of the SRG.