  We study a $PT$-symmetric quantum mechanical model with an O(N)-symmetric
potential of the form $m^{2}\vec{x}^{2}/2-g(\vec{x}^{2})^{2}/N$ using its
equivalent Hermitian form. Although the corresponding classical model has
finite-energy trajectories that escape to infinity, the spectrum of the quantum
theory is proven to consist only of bound states for all $N$. We show that the
model has two distinct phases in the large-$N$ limit, with different scaling
behaviors as $N$ goes to infinity. The two phases are separated by a
first-order phase transition at a critical value of the dimensionless parameter
$m^{2}/g^{2/3}$, given by $3\cdot2^{1/3}$.
