Applying the linear delta expansion to `i phi^3'

TL;DR

This paper applies the linear delta expansion to a non-Hermitian Hamiltonian with an imaginary cubic term, providing proofs of convergence in zero dimensions and numerical evidence in quantum mechanics.

Contribution

It extends the linear delta expansion method to non-Hermitian Hamiltonians with complex potentials, demonstrating convergence properties.

Findings

Proof of convergence in zero-dimensional case

Numerical evidence of convergence in quantum mechanics

Real, positive spectrum for the non-Hermitian Hamiltonian

Abstract

The linear $\delta$ expansion (LDE) is applied to the Hamiltonian $H=(p^2 +m^2 x^2)/2 + igx^3$, which arises in the study of Lee-Yang zeros in statistical mechanics. Despite being non-Hermitian, this Hamiltonian appears to possess a real, positive spectrum. In the LDE, as in perturbation theory, the eigenvalues are naturally real, so a proof of this property devolves on the convergence of the expansion. A proof of convergence of a modified version of the LDE is provided for the $ix^3$ potential in zero dimensions. The methods developed in zero dimensions are then extended to quantum mechanics, where we provide numerical evidence for convergence.