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Non-Hermitian quantum physics is used successfully for the description of different puzzling experimental results, which are observed in open quantum systems. Mostly, the influence of exceptional points on the dynamical properties of the system is studied. At these points, two complex eigenvalues

The properties of open quantum systems are described usually by averaging over the spectroscopic features of the individual states of the system. Open quantum systems are characterized by, e.g., their Markovian and non-Markovian behavior or by equilibrium and nonequilibrium properties. The present-day high-resolution experimental studies provide however more information, namely, concrete information on the spectroscopic properties of the individual states. It is a challenge for the theory to describe these results.

In many current theoretical studies, a non-Hermitian formalism is used for the description of individual states of an open quantum system. In these papers, the part of the system, which is localized in a certain finite space region, is considered to be embedded into an infinitely extended environment. Mathematically, the total function space consists of two parts: the localized part of the system and the extended environment. Mostly, the environment is assumed to be the continuum of scattering wavefunctions^{1}

This formalism is, from the point of view of mathematics, much more complicated than the familiar Hermitian formalism. For example, the eigenfunctions of a non-Hermitian operator

In mathematics, the so-called exceptional points (EPs) are known for a long time; see the book [

Due to the important role of EPs in non-Hermitian systems, in most theoretical studies of non-Hermitian quantum physics only the eigenvalues of the non-Hermitian Hamiltonian

The meaning of the eigenvalues and that of the eigenfunctions of

Further studies in non-Hermitian quantum physics have shown the following nontrivial results. The coupling between system and environment may occur via an exchange of particles or of information (mostly in terms of excitons). We have the following in detail.

(i) Every state

(ii) Every state of the system is coupled exclusively to only one channel with, respectively, some gain from the environment and some loss to the environment. This case is realized, e.g., in the photosynthesis in which visible light is captured in the light-harvesting complex [

Another problem of non-Hermitian quantum physics is the following. In mathematics, the EPs are defined in relation to one environment (mostly called channel). Physical systems are related however generally to more than one channel. For example, transmission of particles through a system needs at least two channels, entrance and exit channel. In this case, the different channels are parts of the total environment and are orthogonal to one another. Another example is the above-mentioned processes with gain and loss each of which occurs relative to another channel. The role played by EPs in physical systems is therefore not at all clear.

Consideration of the eigenfunctions of the non-Hermitian Hamiltonian

In the present paper, we will illustrate the relation between the eigenvalues ^{2}

For this aim, we start from a system with two states

The paper is organized in the following manner. In Section

To begin with, we sketch the features typical for an open quantum system embedded in one common continuum. Details can be found in [

In (

The eigenfunctions of a non-Hermitian Hamilton operator are biorthogonal

Additionally to the Hamiltonian (

The main features characteristic of open quantum systems are described well by the eigenvalues and eigenfunctions of (

The Schrödinger equation

Far from EPs, the coupling of the localized system to the environment influences the spectroscopic properties of the system only marginally [

In the neighborhood of EPs, the coupling between system and environment cause, according to mathematical studies, nonlinear effects in the Schrödinger equation (

Let us consider now the genuine

Also the eigenvalues

The two environments are different from and orthogonal to one another. Further, the two eigenstates with equal

We mention here that the Hamiltonian (

Without an EP in the considered parameter range in relation to both channels, we have

In analogy to (

Using (

We repeat here that, according to their definition [

The resonance structure of the

According to the results obtained in [

The one-channel case does, therefore, not allow us to prove the existence of the nonlinear effects and of EM, since the resonance structure of the cross section calculated with and without EM is the same in this case [

The conservation of the resonance structure of the cross section, which is possible in the one-channel case, is expected to be impossible, generally, in the two-channel (or more-channel) case.

The aim of our numerical studies is first to show the influence of a singularity onto the eigenvalues and eigenfunctions of the non-Hermitian Hamilton operator

In Figure

Eigenvalues

We are interested, above all, in the appearance of a critical parameter value

In Figure

The results are the following. The energies

Around the critical parameter value, the phase rigidity

When we start from parameter independent energies and parameter dependent widths in contrast to the case considered in Figure

We remark that not only do eigenstates with positive widths appear in the present study on systems embedded in two environments but also they are well known from different studies on one-channel systems.

Observable information on the spectroscopic properties of the localized part of the system is contained in the resonance structure of the cross section. Using (

In all cases we see the double-hump structure of the transmission which is characteristic of the resonance structure of a two-level system coupled to one channel [

Resonance structure (above) and contour plot (below) of the transmission with

Additionally, we have performed some calculations with different values of the original widths (

When the EM of the eigenstates via the continuum of scattering wavefunctions is different for the two channels and different from zero,

The same as Figure

Although the double-hump structure of the cross section, appearing under the condition

The same as Figure

Further information is contained in the contour plots of the cross section some of which are shown in the lower parts of Figures

In all cases, the two eigenstates of the non-Hermitian operator

The results of our calculations show very clearly that not only the eigenvalues

While the influence of the eigenvalues is restricted, above all, to a small parameter range around some critical points, the eigenfunctions influence a much larger parameter range around these points. An example is the EM (which is a second-order effect). Its influence cannot be neglected over a comparably large parameter range. This behavior is known from the one-channel case [

Instead of an EP, we see in Figure

Due to the width bifurcation, the width of the state with originally vanishing coupling strength to one of the channels becomes automatically positive. That means that this state gains something from the environment. Thus some gain from the environment in an open quantum system is not at all an exotic process. The different characteristic features of non-Hermitian quantum physics, which are considered in the Introduction, are really nothing but different sides of non-Hermitian quantum physics.

In conclusion we state the following. According to long-time experience, open quantum systems are described usually well by standard methods. There are however some exceptions which occur in a parameter range around singularities. These may be the well-known exceptional points (EPs). Others are related to the fact that the biorthogonal eigenfunctions of

In any case, the non-Hermitian formalism is a very powerful method and is able to explain different phenomena that are puzzling in standard Hermitian quantum physics. There remain however many open questions to which an answer has to be found in future. One of these questions is the mathematical and physical meaning of the critical point

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors thank Jon Bird for valuable discussions.

We consider open quantum systems described by a non-Hermitian Hamilton operator. This should not be confused with the consideration of PT-symmetric systems which are neither open nor closed, but nonisolated according to the definition in, e.g., C.M. Bender, Journal of Physics: Conference Series 631, 012002 (2015).

In contrast to the definition that is used in, for example, nuclear physics, we define the complex energies before and after diagonalization of