Nonunique C operator in PT Quantum Mechanics

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 $H=1/2p^2+1/2{\mu }^2{q}^2+i\epsilon q^3$ 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.

10 pages, 1 figure

References in corpus (1)

• Dual oscillators and Quantum Pendulums: spectrum and correlators

Cited by in corpus (2)

• Fundamental length in quantum theories with PT-symmetric Hamiltonians
• Small-$\epsilon$ behavior of the Non-Hermitian PT-Symmetric Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$