Generating Converging Bounds to the (Complex) Discrete States of the $P^2 + iX^3 + i\alpha X$ Hamiltonian

TL;DR

This paper applies the Eigenvalue Moment Method to a complex Hamiltonian, generating converging bounds for its complex energy eigenvalues, thus advancing numerical techniques for non-Hermitian quantum systems.

Contribution

It introduces a novel application of the Eigenvalue Moment Method to a complex Hamiltonian, providing a systematic way to bound its complex eigenvalues.

Findings

Successfully generated converging bounds for complex energies.

Validated the method against asymptotic predictions.

Enhanced numerical tools for non-Hermitian quantum mechanics.

Abstract

The Eigenvalue Moment Method (EMM), Handy (2001), Handy and Wang (2001)) is applied to the $H_\alpha \equiv P^2 + iX^3 + i\alpha X$ Hamiltonian, enabling the algebraic/numerical generation of converging bounds to the complex energies of the $L^2$ states, as argued (through asymptotic methods) by Delabaere and Trinh (J. Phys. A: Math. Gen. {\bf 33} 8771 (2000)).