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 $3\cdot2^{ 1/3}$.
Categories: hep-th math-ph math.MP quant-ph