Universal spectral behavior of $x^2(ix)^ε$ potentials

Abstract

The PT-symmetric Hamiltonian $H=p^2+x^2(ix)^ε$ ($ε$ real) exhibits a phase transition at $ε=0$. When $ε\geq0$, the eigenvalues are all real, positive, discrete, and grow as $ε$ increases. However, when $ε<0$ there are only a finite number of real eigenvalues. As $ε$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $ε=-1$. 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 $H^{(2n)}=p^{2n}+x^2(ix)^ε$ ($ε$ real, n=1, 2, 3, ...). The complex classical behaviors of these Hamiltonians are also examined.