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

TL;DR

This paper demonstrates that the spectral properties of a class of PT-symmetric Hamiltonians, including the well-known $x^2(ix)^psilon$, are universal across different powers, revealing consistent phase transition behaviors and complex classical dynamics.

Contribution

It generalizes the spectral behavior observed in $x^2(ix)^psilon$ to a broader class of Hamiltonians $H^{(2n)}=p^{2n}+x^2(ix)^psilon$, establishing their universal spectral characteristics.

Findings

Spectral behaviors are universal across the class of Hamiltonians.

Eigenvalues are real and positive for psilon0, finite for psilon<0.

Number of real eigenvalues decreases to one as psilon approaches -1.

Abstract

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.