Title: Eigenvalue and Eigenfunction for the $PT$-symmetric Potential $V = - (ix)^N$,
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?