Dirichlet spectra of the paradigm model of complex PT-symmetric potential: V(x)=−(ix)N

So far the spectra En(N) of the paradigm model of complex PT(Parity-Time)-symmetric potential VBB(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(λ) is an analytic function of a real parameter λ, its eigenvalues En(λ) may not be analytic at finite number of Isolated Points (IPs). In this light, we present the Dirichlet spectra En(N) of VBB(x,N) for 2≤N<12 using the numerical integration of Schrödinger equation with ψ(x=±∞)=0 and the diagonalization of H=p2∕2μ+VBB(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∈R] such that |a| is the largest for a given energy among all (multiple) turning points. We find that En(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 VBB becomes a Hermitian flat-top potential barrier). At N=6 and 10, VBB(x,N) becomes a Hermitian well and we recover its well known eigenvalues.