Non-Hermitian Hamilton operator in open quantum systems

TL;DR

This paper discusses the non-Hermitian Hamiltonian in open quantum systems using the Feshbach projection operator formalism, highlighting phenomena like resonance states, bound states in the continuum, and effects of branch points in the complex plane.

Contribution

It provides a detailed analysis of the FPO method's peculiarities and illustrates key phenomena such as decay, scattering, and spectroscopic reordering in open quantum systems.

Findings

Unified description of decay and scattering processes

Identification of bound states in the continuum

Spectroscopic reordering in overlapping resonances

Abstract

In the Feshbach projection operator (FPO) formalism the whole function space is divided into two subspaces. One of them contains the wave functions localized in a certain finite region while the continuum of extended scattering wave functions is involved in the other subspace. The Hamilton operator of the whole system is Hermitian, that of the localized part is, however, non-Hermitian. This non-Hermitian Hamilton operator $H_{\rm eff}$ represents the core of the FPO method in present-day studies. It gives a unified description of discrete and resonance states. Furthermore, it contains the time operator. The eigenvalues $z_\lambda$ and eigenfunctions $\phi_\lambda$ of $H_{\rm eff}$ are an important ingredient of the $S$ matrix. They are energy dependent. The phases of the $\phi_\lambda$ are, generally, nonrigid. Most interesting physical effects are caused by the branch points in the complex plane. On the one hand, they cause the avoided level crossings that appear as level repulsion or widths bifurcation in approaching the branch points under different conditions. On the other hand, observable values are usually enhanced and accelerated in the vicinity of the branch points. In most cases, the theory is time asymmetric. An exception are the ${\cal PT}$ symmetric bound states in the continuum appearing in space symmetric systems due to the avoided level crossing phenomenon in the complex plane. In the paper, the peculiarities of the FPO method are considered and three typical phenomena are sketched: (i) the unified description of decay and scattering processes, (ii) the appearance of bound states in the continuum and (iii) the spectroscopic reordering processes characteristic of the regime with overlapping resonances.