Multiscale reference function analysis of the PT symmetry breaking solutions for the P-2+iX(3)+i alpha X Hamiltonian

The recent work of Delabaere and Trinh (Delabaere E and Trinh D T 2000 J. P hys. A: Math. Gen. 33 8771) discovered the existence of PT symmetry breakin g, complex energy, L-2 solutions for the one-dimensional Hamiltonian, P-2 iX(3) + i alphaX, in the asymptotic limit alpha --> -infinity. Their asymp totic analysis produced questionable results for moderate values of alpha. We can easily confirm the existence of PT symmetry breaking solutions by ex plicitly computing the low-lying states for /alpha/ < O(10). Our analysis m akes use of the multiscale reference function (MRF) approach, developed by Tymczak et al (Tymczak C J, Japaridze G S, Handy C R and Wang Xiao-Qian 199 8a Phys. Rev. Lett. 80 3678; 1998b Phys. Rev. A 58 2708). The MRF results c an be validated by comparing them with the converging eigenenergy bounds ge nerated through the eigenvalue moment method, as recently argued by Handy ( 2001a, b). Given the reliability of the MRF analysis, its fast numerical im plementation, high accuracy and theoretical simplicity, the present formali sm defines an effective and efficient procedure for analysing many related problems that have appeared in the recent literature.