Interactions of Hermitian and non-Hermitian Hamiltonians

TL;DR

This paper investigates how coupling Hermitian and non-Hermitian PT-symmetric Hamiltonians affects their energy spectra, showing that energies remain real under weak coupling but become complex beyond a critical strength.

Contribution

It provides a detailed analysis of the spectral behavior when Hermitian and non-Hermitian Hamiltonians are coupled, including perturbative results for ground-state energies.

Findings

Energy remains real for small coupling strengths.

Energy becomes complex beyond a critical coupling value.

Ground-state energy is real up to second-order perturbation theory.

Abstract

The coupling of non-Hermitian PT-symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy remains real for small values of the coupling constant, but becomes complex if the coupling becomes stronger than some critical value. For a quadratic non-Hermitian PT-symmetric Hamiltonian coupled to an arbitrary real Hermitian PT-symmetric Hamiltonian, the reality of the ground-state energy for small enough coupling constant is established up to second order in perturbation theory.