Small-$\epsilon$ behavior of the Non-Hermitian PT-Symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$

TL;DR

This paper investigates the small-epsilon expansion of the C operator and the isospectral Hermitian Hamiltonian for a class of PT-symmetric Hamiltonians, revealing new perturbative features beyond eigenvalues.

Contribution

It derives the small-epsilon expansion of the C operator and the equivalent Hermitian Hamiltonian, providing new insights into the structure of PT-symmetric systems.

Findings

Derived the small-epsilon expansion of the C operator

Obtained the isospectral Dirac-Hermitian Hamiltonian

Revealed new features of the Hamiltonian beyond eigenvalues

Abstract

The energy eigenvalues of the class of non-Hermitian PT-symmetric Hamiltonians $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) are real, positive, and discrete. The behavior of these eigenvalues has been studied perturbatively for small $\epsilon$. However, until now no other features of $H$ have been examined perturbatively. In this paper the small-$\epsilon$ expansion of the C operator and the equivalent isospectral Dirac-Hermitian Hamiltonian $h$ are derived.