An existence criterion for the PT -symmetric phase transition

We consider on $L^2(\R)$ the Schroedinger operator family $H(g)$ with domain and action defined as follows $$ D(H(g))=H^2(\R)\cap L^2_{2M}(\R); \quad H(g) u=\bigg(-\frac{d^2}{dx^2}+\frac{x^{2M}}{2M}-g\,\frac{x^{M-1}}{M-1}\bigg)u $$ where $g\in\C$, $M=2,4,\ldots\;$. $H(g)$ is self-adjoint if $g\in\R$, while $H(ig)$ is $\PT$-symmetric. We prove that $H(ig)$ exhibits the so-called $\P\T$-symmetric phase transition. Namely, for each eigenvalue $E_n(ig)$ of $H(ig)$, $g\in\R$, there exist $R_1(n)>R(n)>0$ such that $E_n(ig)\in\R$ for $|g|<R(n)$ and turns into a pair of complex conjugate eigenvalues for $|g|>R_1(n)$.