Analytic approximation for eigenvalues of a class of $\mathcal{PT}$ symmetric Hamiltonians

TL;DR

This paper develops an analytical approximation method for calculating eigenvalues of a class of $ ext{PT}$-symmetric Hamiltonians, achieving high accuracy across all energy levels and parameter ranges.

Contribution

It introduces a new approximation technique using harmonic-oscillator basis functions with variable parameters for $ ext{PT}$-symmetric Hamiltonians.

Findings

High accuracy eigenvalue approximations for all energy levels.

Effective across all $ ext{PT}$-symmetric Hamiltonian parameters.

Applicable to a broad class of $ ext{PT}$-symmetric systems.

Abstract

An analytical approximation for the eigenvalues of $\mathcal{PT}$ symmetric Hamiltonian $\mathsf{H} = -d^{2}/dx^{2} - (\mathrm{i}x)^{\epsilon+2}$, $\epsilon > -1$ is developed via simple basis sets of harmonic-oscillator wave functions with variable frequencies and equilibrium positions. We demonstrate that our approximation provides high accuracy for any given energy level for all values of $\epsilon > -1$.