Multiscale reference function analysis of the 𝒫𝒯 symmetry breaking solutions for the<i>P</i>²+ i<i>X</i>³+ iα<i>X</i>Hamiltonian

Abstract

The recent work of Delabaere and Trinh (Delabaere E and Trinh D T 2000 J. Phys. A: Math. Gen. 33 8771) discovered the existence of symmetry breaking, complex energy, L2 solutions for the one-dimensional Hamiltonian, P2 + iX 3 + iαX, in the asymptotic limit α→-∞. Their asymptotic analysis produced questionable results for moderate values of α. We can easily confirm the existence of symmetry breaking solutions by explicitly computing the low-lying states for |α|<O(10). Our analysis makes 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 1998a Phys. Rev. Lett. 80 3678; 1998b Phys. Rev. A 58 2708). The MRF results can be validated by comparing them with the converging eigenenergy bounds generated through the eigenvalue moment method, as recently argued by Handy (2001a, b). Given the reliability of the MRF analysis, its fast numerical implementation, high accuracy and theoretical simplicity, the present formalism defines an effective and efficient procedure for analysing many related problems that have appeared in the recent literature.