%0 Journal Article
%T Universal spectral behavior of $x^2(ix)^ŠĆ$ potentials
%A Carl M. Bender
%A Daniel W. Hook
%J Physics
%D 2012
%I arXiv
%R 10.1103/PhysRevA.86.022113
%X The PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon$ real) exhibits a phase transition at $\epsilon=0$. When $\epsilon\geq0$, the eigenvalues are all real, positive, discrete, and grow as $\epsilon$ increases. However, when $\epsilon<0$ there are only a finite number of real eigenvalues. As $\epsilon$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $\epsilon=-1$. 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 $H^{(2n)}=p^{2n}+x^2(ix)^\epsilon$ ($\epsilon$ real, n=1, 2, 3, ...). The complex classical behaviors of these Hamiltonians are also examined.
%U http://arxiv.org/abs/1205.4425v1