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 $H(\hat{I}»)$ is an analytic function of a real parameter $\hat{I}»$, its eigenvalues $E_n(\hat{I}»)$ 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{ö}dinger equation with $\ddot{I}ˆ(x=\pm \infty )=0$ and the diagonalization of $H=p^2/2\hat{I}¼+V_{BB}(x,N)$ 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) [$-a+ib,a+ib,a,b\in \mathcal{R}$] 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.

Final Version just appeared in Annals of Physics (N.Y.) 383 (2017) 635-644