Vector Models in PT Quantum Mechanics

TL;DR

This paper explores non-Hermitian Hamiltonians with vector perturbations within PT quantum mechanics, analyzing algebraic structures like E3 and SO(3), and identifying spectral transition points.

Contribution

It extends PT quantum mechanics to include vector perturbations in E3 and SO(3) algebras, and applies the Wigner-Eckart theorem in a non-Hermitian context.

Findings

Identified critical coupling values for spectral transitions.

Generalized E2 algebra to E3 in PT quantum mechanics.

Extended Wigner-Eckart theorem to non-Hermitian models.

Abstract

We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of PT quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.