  We study a simple class of unitary renormalization group transformations
governed by a parameter f in the range [0,1]. For f=0, the transformation is
one introduced by Wegner in condensed matter physics, and for f=1 it is a
simpler transformation that is being used in nuclear theory. The transformation
with f=0 diagonalizes the Hamiltonian but in the transformations with f near 1
divergent couplings arise as bound state thresholds emerge. To illustrate and
diagnose this behavior, we numerically study Hamiltonian flows in two simple
models with bound states: one with asymptotic freedom and a related one with a
limit cycle. The f=0 transformation places bound-state eigenvalues on the
diagonal at their natural scale, after which the bound states decouple from the
dynamics at much smaller momentum scales. At the other extreme, the f=1
transformation tries to move bound-state eigenvalues to the part of the
diagonal corresponding to the lowest momentum scales available and inevitably
diverges when this scale is taken to zero. We analyze the shift mechanism
analytically in a 3x3 matrix model, which displays the essense of this RG
behavior.
