Generating Converging Eigenenergy Bounds for the Discrete States of the -ix^3 Non-Hermitian Potential

TL;DR

This paper introduces a novel method to generate converging bounds for the discrete eigenenergies of the non-Hermitian potential -ix^3, supporting the hypothesis that its spectrum is real, through transforming the Schrödinger equation into a fourth order differential equation.

Contribution

The authors develop a new formulation transforming the Schrödinger equation into a fourth order differential equation for |Psi(x)|^2, enabling the application of the Eigenvalue Moment Method to bound energies of the -ix^3 potential.

Findings

Generated tight bounds for the first five discrete energy levels.

Provided evidence supporting the reality of the spectrum for the -ix^3 potential.

Demonstrated rapid convergence of bounds for low-lying states.

Abstract

Recent investigations by Bender and Boettcher (Phys. Rev. Lett 80, 5243 (1998)) and Mezincescu (J. Phys. A. 33, 4911 (2000)) have argued that the discrete spectrum of the non-hermitian potential $V(x) = -ix^3$ should be real. We give further evidence for this through a novel formulation which transforms the general one dimensional Schrodinger equation (with complex potential) into a fourth order linear differential equation for $|\Psi(x)|^2$. This permits the application of the Eigenvalue Moment Method, developed by Handy, Bessis, and coworkers (Phys. Rev. Lett. 55, 931 (1985);60, 253 (1988a,b)), yielding rapidly converging lower and upper bounds to the low lying discrete state energies. We adapt this formalism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.