Asymptotic freedom and bound states in Hamiltonian dynamics

We study a model of asymptotically free theories with bound states using the similarity renormalization group for Hamiltonians. We find that the renormalized effective Hamiltonians can be approximated in a large range of widths by introducing similarity factors and the running coupling constant. This approximation loses accuracy for the small widths on the order of the bound state energy and it is improved by using the expansion in powers of the running coupling constant. The coupling constant for small widths is of order 1. The small width effective Hamiltonian is projected on a small subset of the effective basis states. The resulting small matrix is diagonalized exactly and the bound state energy of the original theory is obtained with accuracy of the order of 10% using the first three terms in the perturbative expansion of the effective Hamiltonian. We briefly describe options for improving the accuracy.