  We present exact tree-order amplitudes for $H^* \to n~H$, for final states
containing one or two particles with non-zero three-momentum, for various
interaction potentials. We show that there are potentials leading to tree
amplitudes that satisfy unitarity, not only at threshold but also in the above
kinematical configurations and probably beyond. As a by-product, we also
calculate $2\to n$ tree amplitudes at threshold and show that for the unbroken
$\phi^4$ theory they vanish for $n>4~$, for the Standard Model Higgs they
vanish for $n\ge 3~$ and for a model potential, respecting tree-order
unitarity, for $n$ even and $n>4~$. Finally, we calculate the imaginary part of
the one-loop $1\to n$ amplitude in both symmetric and spontaneously broken
$\phi^4$ theory.
