Calculations of high multiplicity Higgs amplitudes exhibit a rapid growth that may signal an end of perturbative behavior or even the need for new physics phenomena. As a step towards this problem we consider the quantum mechanical equivalent of $1 to n$ scattering amplitudes in a spontaneously broken $phi^4$-theory by extending our previous results on the quartic oscillator with a single minimum to transitions $langle n lvert hat{x} rvert 0 rangle$ in the symmetric double-well potential with quartic coupling $lambda$. Using recursive techniques to high order in perturbation theory, we argue that these transitions are of exponential form $langle n lvert hat{x} rvert 0 rangle sim exp left( F (lambda n) / lambda right)$ in the limit of large $n$ and $lambda n$ fixed. We apply the methods of exact perturbation theory put forward by Serone et al. to obtain the exponent $F$ and investigate its structure in the regime where tree-level perturbation theory violates unitarity constraints. We find that the resummed exponent is in agreement with unitarity and rigorous bounds derived by Bachas.
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