# Eigenvalue and Eigenfunction for the $PT$-symmetric Potential $V = - (ix)^N$

## TL;DR

This paper investigates the eigenvalues and eigenfunctions of the PT-symmetric Hamiltonian V = - (ix)^N, exploring path dependence, eigenfunction shape, and the structure of multiple real spectra for large N.

## Contribution

It extends previous work by analyzing how different integration paths affect eigenvalues and eigenfunctions, and examines the structure of multiple spectra for large N.

## Key findings

- Eigenvalues can be obtained along various paths, not just the central Stokes wedge.
- Eigenfunctions' shapes may depend on the integration path.
- Multiple real spectra emerge for large N, with potential relations among them.

## Abstract

The real energy spectrum from the $PT$-symmetric Hamiltonian $H = p^2 - (ix)^N$ with $x\in\mathbb{C}$ was examined within one pair of Stokes wedges in 1998 by Bender and Boettcher. For this Hamiltonian we discuss the following three questions. First, since their paper used a Runge-Kutta method to integrate along a path at the center of the Stokes wedges to calculate eigenvalues $E$ with high accuracy, we wonder if the same eigenvalues can be obtained if integrate along some other paths in different shapes. Second, what the corresponding eigenfunctions look like? Should the eigenfunctions be independent from the shapes of path or not? Third, since for large $N$ the Hamiltonian contains many pairs of Stokes wedges symmetric with respect to the imaginary axis of $x$, thus multiple families of real energy spectrum can be obtained. What do they look like? Any relation among them?