Multiscale Reference Function Analysis of the ${\cal P}{\cal T}$ Symmetry Breaking Solutions for the $P^2+iX^3+i\alpha X$ Hamiltonian

TL;DR

This paper employs the Multiscale Reference Function approach to analyze ${ m PT}$-symmetry breaking solutions in a complex Hamiltonian, confirming their existence for moderate parameter values with high accuracy and efficiency.

Contribution

It introduces a reliable, fast, and simple MRF-based method for analyzing ${ m PT}$-symmetry breaking solutions in complex Hamiltonians, validated against eigenenergy bounds.

Findings

Confirmed ${ m PT}$-symmetry breaking solutions for moderate $oldsymbol{ extit{\alpha}}$ values.

Validated MRF results with converging eigenenergy bounds.

Demonstrated the effectiveness of MRF in analyzing complex quantum systems.

Abstract

The recent work of Delabaere and Trinh (2000 J. Phys. A 33 8771) discovered the existence of ${\cal P}{\cal T}$-symmetry breaking, complex energy, $L^2$ solutions for the one dimensional Hamiltonian, $P^2+iX^3+i\alpha X$, in the asymptotic limit, $\alpha \to -\infty$. Their asymptotic analysis produced questionable results for moderate values of $\alpha$. We can easily confirm the existence of ${\cal P}{\cal T}$-symmetry breaking solutions, by explicitly computing the low lying states, for $|\alpha| < O (10)$. Our analysis makes use of the Multiscale Reference Function (MRF) approach, developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3678; 1998 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 analyzing many related problems that have appeared in the recent literature.