  Calculations of $1\to N$ amplitudes in scalar field theories at very high
multiplicities exhibit an extremely rapid growth with the number $N$ of final
state particles. This either indicates an end of perturbative behaviour, or
possibly even a breakdown of the theory itself. It has recently been proposed
that in the Standard Model this could even lead to a solution of the hierarchy
problem in the form of a "Higgsplosion". To shed light on this question we
consider the quantum mechanical analogue of the scattering amplitude for $N$
particle production in $\phi^4$ scalar quantum field theory, which corresponds
to transitions $\langle N \lvert \hat{x} \rvert 0 \rangle$ in the anharmonic
oscillator with quartic coupling $\lambda$. We use recursion relations to
calculate the $\langle N \lvert \hat{x} \rvert 0 \rangle$ amplitudes to high
order in perturbation theory. Using this we provide evidence that the amplitude
can be written as $\langle N \lvert \hat{x} \rvert 0 \rangle \sim
\exp(F(\lambda N)/\lambda)$ in the limit of large $N$ and $\lambda N$ fixed. We
go beyond the leading order and provide a systematic expansion in powers of
$1/N$. We then resum the perturbative results and investigate the behaviour of
the amplitude in the region where tree-level perturbation theory violates
unitarity constraints. The resummed amplitudes are in line with unitarity as
well as stronger constraints derived by Bachas. We generalize our result to
arbitrary states and powers of local operators $\langle N \lvert \hat{x}^q
\rvert M \rangle$ and confirm that, to exponential accuracy, amplitudes in the
large $N$ limit are independent of the explicit form of the local operator,
i.e. in our case $q$.
