The $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H=p^2+x^2{\left(ix\right)}^{\varepsilon }$ ($\varepsilon$ real) exhibits a phase transition at $\varepsilon =0$. When $\varepsilon ⩾0$, the eigenvalues are all real, positive, discrete, and grow as $\varepsilon$ increases. However, when $\varepsilon <0$ there are only a finite number of real eigenvalues. As $\varepsilon$ approaches $-1$ from above, the number of real eigenvalues decreases to 1, and this eigenvalue becomes infinite at $\varepsilon =-1$. In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians $H^{\left(2n\right)}=p^{2n}+x^2{\left(ix\right)}^{\varepsilon }$ ($\varepsilon$ real, $n=1,2,3,...$). The complex classical behaviors of these Hamiltonians are also examined.

• Received 25 May 2012

DOI:https://doi.org/10.1103/PhysRevA.86.022113