# Behavior of eigenvalues in a region of broken-PT symmetry

## TL;DR

This paper investigates the eigenvalue behavior of PT-symmetric quantum Hamiltonians in the broken symmetry region, revealing new phenomena such as an infinite-order exceptional point and spectral transitions.

## Contribution

It provides the first detailed numerical and analytical analysis of eigenvalues in the broken PT symmetry region for <, uncovering novel spectral transitions and critical points.

## Key findings

- Discovery of an infinite-order exceptional point at 
- Transition from discrete to partially continuous spectrum at 
- Eigenvalue behavior near the conformal limit 

## Abstract

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. When $\varepsilon\geq0$, the eigenvalues of this non-Hermitian Hamiltonian are discrete, real, and positive. This portion of parameter space is known as the region of unbroken PT symmetry. In the region of broken PT symmetry $\varepsilon<0$ only a finite number of eigenvalues are real and the remaining eigenvalues appear as complex-conjugate pairs. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry has thus far been unexplored. This paper presents a detailed numerical and analytical examination of the behavior of the eigenvalues for $-4<\varepsilon<0$. In particular, it reports the discovery of an infinite-order exceptional point at $\varepsilon=-1$, a transition from a discrete spectrum to a partially continuous spectrum at $\varepsilon=-2$, a transition at the Coulomb value $\varepsilon=-3$, and the behavior of the eigenvalues as $\varepsilon$ approaches the conformal limit $\varepsilon=-4$.