Calculations of $1to N$ 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 $q$.
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