Nonunique C operator in PT Quantum Mechanics

TL;DR

This paper demonstrates that the $C$ operator in PT-symmetric quantum mechanics is nonunique, with an infinite set of solutions, each leading to a different equivalent Hermitian Hamiltonian.

Contribution

It shows that the algebraic equations defining the $C$ operator admit infinitely many solutions, revealing nonuniqueness in PT quantum mechanics.

Findings

The $C$ operator for a specific PT Hamiltonian is determined perturbatively.

The $C$ operator contains an infinite number of arbitrary parameters.

Different $C$ operators lead to different equivalent Hermitian Hamiltonians.

Abstract

The three simultaneous algebraic equations, $C^2=1$, $[C,PT]=0$, $[C,H]=0$, which determine the $C$ operator for a non-Hermitian $PT$-symmetric Hamiltonian $H$, are shown to have a nonunique solution. Specifically, the $C$ operator for the Hamiltonian $H={1/2}p^2+{1/2}\mu^2q^2+i\epsilon q^3$ is determined perturbatively to first order in $\epsilon$ and it is demonstrated that the $C$ operator contains an infinite number of arbitrary parameters. For each different $C$ operator, the corresponding equivalent isospectral Dirac-Hermitian Hamiltonian $h$ is calculated.