Dirichlet spectrum of the paradigm model of complex PT-symmetric potential: $V(x)=-(ix)^N$

Abstract: So far the spectra $E_n(N)$ of the paradigm model of complex PT(Parity-Time)-symmetric potential $V_{BB}(x,N)=-(ix)^N$ is known to be analytically continued for $N > 4$. Consequently, the well known eigenvalues of the Hermitian cases ($N=6,10$) cannot be recovered. Here, we illustrate Kato's theorem that even if a Hamiltonian $H(\lambda)$ is an analytic function of a real parameter $\lambda$, its eigenvalues $E_n(\lambda)$ may not be analytic at finite number of Isolated Points (IPs). In this light, we present the Dirichlet spectra $E_n(N)$ of $V_{BB}(x,N)$ for $2\le N<12$ using the numerical integration of Schr{\"o}dinger equation with $\psi(x=\pm \infty)=0$ and the diagonalization of $H=p^2/2\mu+V_{BB}(x,N)$ in the harmonic oscillator basis. We show that these real discrete spectra are consistent with the most simple two-turning point CWKB (C refers to complex turning points) method provided we choose the maximal turning points (MxTP) [$-a+ib,a+ib, a, b \in {\cal R}$] such that $|a|$ is the largest for a given energy among all (multiple) turning points. We find that $E_n(N)$ are continuous function of $N$ but non-analytic (their first derivative is discontinuous) at IPs $N=4,8$; where the Dirichlet spectrum is null (as $V_{BB}$ becomes a Hermitian flat-top potential barrier). At $N=6$ and $10$, $V_{BB}(x,N)$ becomes a Hermitian well and we recover its well known eigenvalues.