Research Article | Open Access

Hichem Eleuch, Ingrid Rotter, "Loss, Gain, and Singular Points in Open Quantum Systems", *Advances in Mathematical Physics*, vol. 2018, Article ID 3653851, 9 pages, 2018. https://doi.org/10.1155/2018/3653851

# Loss, Gain, and Singular Points in Open Quantum Systems

**Academic Editor:**Soheil Salahshour

#### Abstract

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 of the non-Hermitian Hamiltonian coalesce (where is the energy and is the inverse lifetime of the state ). We show that also the eigenfunctions of the two states play an important role, sometimes even the dominant one. Besides exceptional points, other critical points exist in non-Hermitian quantum physics. At these points in the parameter space, the biorthogonal eigenfunctions of become orthogonal. For illustration, we show characteristic numerical results.

#### 1. Introduction

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 are biorthogonal with nontrivial consequences for their normalization. Most important is however the existence of singularities.

In mathematics, the so-called exceptional points (EPs) are known for a long time; see the book [1] on the perturbation theory for linear operators. At the EPs, two eigenvalues of coalesce. This coalescence is, on principle, different from the well-known degeneration of two eigenstates of a Hermitian operator . In the last case, the eigenfunctions of the two states are different and orthogonal while those of two coalescing eigenstates of are biorthogonal and differ only by a phase from one another at the EP. The related interesting topological features are studied also experimentally in detail [2–4].

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 are considered in detail. The eigenfunctions are often assumed to be of minor interest. Examples are the method of complex scaling [5] and the review [6] on superradiance. The meaning of the eigenvalues as well as of the eigenfunctions of is discussed theoretically in the review [7] and compared to experimental results in the review [8]. The eigenfunctions around EPs play an important role for the explanation of some experimental results that are puzzling from the point of view of Hermitian quantum physics. Examples are the so-called phase lapses [9] observed experimentally in mesoscopic systems and the high efficiency of the photosynthesis [10]. The role of EPs in quantum physics is discussed in many other papers [11–16].

The meaning of the eigenvalues and that of the eigenfunctions of are less different from one another than it seems to be. On the one hand, the eigenfunctions of play a role also in those cases in which they are usually not considered [7, 8]. Among others, they are responsible for the fact that the influence of the EPs occurs in a larger parameter range around the EP than expected from the behavior of the eigenvalues. Moreover, higher-order EPs are shielded from observation in physical systems [17] because of the mixing of the eigenfunctions of via the environment. On the other hand, the eigenvalues of determine the position of the EPs and influence therefore also the behavior 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 of the system is related to different decay channels according to the values of the so-called partial widths of the state relative to the channels . The coupling of the whole system to the environment is characterized by the sum of partial widths. This case is realized in many-body systems. In standard Hermitian quantum physics, the exchange of particles between system and environment is described as tunneling of particles.

(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 [10]. The exciton transfer occurs without excitation of any eigenstates of . It cannot be described in standard theory.

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 detail has shown further nontrivial properties of non-Hermitian quantum physics. For example, the EPs are not the only singular points in non-Hermitian quantum systems. Other singularities occur at maximum width bifurcation or level repulsion where the eigenfunctions of are orthogonal (instead of biorthogonal). They are related to obviously nonlinear irreversible processes [17].

In the present paper, we will illustrate the relation between the eigenvalues ^{2} and eigenfunctions of the non-Hermitian Hamiltonian , on the one hand, as well as the appearance of gain and loss and different types of singularities in non-Hermitian quantum physics, on the other hand.

For this aim, we start from a system with two states the complex energies of which are . We then consider the coupling of the system to two channels under the assumption that each state is coupled, above all, to only one channel: . We vary parametrically the energies of the states and determine the eigenvalues of the non-Hermitian Hamiltonian . In such a case, width bifurcation causes automatically a transition from loss (determined by and ) to gain (determined by and ).

The paper is organized in the following manner. In Section 2, we sketch the formalism which is used in performing the calculations. Some numerical results are shown in Section 3. First we show a few eigenvalue trajectories together with the corresponding eigenfunction trajectories in Section 3.1. The resonance structure of the transmission and the transmission probability are given in Section 3.2. In the last Section 4 we summarize and discuss the results and draw some conclusions on non-Hermitian quantum physics of open quantum systems.

#### 2. Non-Hermitian Hamiltonian of an Open Quantum System

##### 2.1. Eigenvalues and Eigenfunctions of

To begin with, we sketch the features typical for an open quantum system embedded in one common continuum. Details can be found in [7] and, above all, in [17]. They can be discussed by means of the genuine non-Hermitian matrix:Here, are the complex energies of the two states which are coupled to the environment (called also channel ). are the energies of the states and are their widths. stand for the coupling matrix elements of the two states via the common environment . They are complex where Re() arises from the principal value integral and Im() from the residuum [7]. The complex eigenvalues of give energies and widths of the states of the localized part of the system.

In (1), a singular point may appear, the so-called EP, at which two eigenvalues of coalesce [1]. At these points, the two corresponding eigenfunctions are not orthogonal. Insteadaccording to analytical and numerical results [18–20]. An EP is, according to its definition, related to the common environment in which the system is embedded. In other words, it is well defined under the condition that the system is embedded in only one continuum.

The eigenfunctions of a non-Hermitian Hamilton operator are biorthogonalIn the case of the symmetric Hamiltonian (1), it isand the eigenfunctions should be normalized according toin order to smoothly describe the transition from a closed system with discrete states to a weakly open one with narrow resonance states. As a consequence of (5), the values of the standard expressions are changed:Furthermore, the phase rigidity, which is a quantitative measure for the biorthogonality of the eigenfunctions,is smaller than 1. Far from an EP, while it approaches the value when an EP is approached.

Additionally to the Hamiltonian (1), we will consider the non-Hermitian matrixwhich describes the system without any mixing of its states via the environment. In other words, corresponds to vanishing EM of the eigenstates. The eigenfunctions of can be represented in the set of eigenfunctions of ,under the condition that are normalized by . The coefficients differ from the . They contain the information on the strength of EM.

The main features characteristic of open quantum systems are described well by the eigenvalues and eigenfunctions of (1). Typical values related to the eigenfunctions are the phase rigidity (9) and the contribution of EM (11) to their purity. All these values contain the influence of the environment. They are proven experimentally; for details see [7, 8, 17].

##### 2.2. Schrödinger Equation with

The Schrödinger equation may be rewritten into a Schrödinger equation with source term [7, 17],In this representation, the coupling of the states and of the localized system via the common environment of scattering wavefunctions (EM) is contained in the source term of the Schrödinger equation; for details see [7].

Far from EPs, the coupling of the localized system to the environment influences the spectroscopic properties of the system only marginally [7, 17]. The influence of the environment is however nonvanishing also in this case; see, e.g., the experimental results [21]. That means, the eigenstates of will never be the same as those of a Hermitian operator.

In the neighborhood of EPs, the coupling between system and environment cause, according to mathematical studies, nonlinear effects in the Schrödinger equation (12). Among others, these effects lead to a conservation of the resonance structure of the cross section in the one-channel case which is finally unaffected by EM and by the existence of EPs [17]. Thus, the one-channel case cannot be used in order to test the results of the non-Hermitian formalism.

##### 2.3. Two-Channel Case with

Let us consider now the genuine non-Hermitian matrixHere, and are the complex energies of the two states relative to the two different channels . Usually . and stand for the coupling matrix elements of the two states via environments and , respectively. The Hamiltonian (13) includes the fact that the complex energy of the localized state may be different relative to the different channels .

Also the eigenvalues and eigenfunctions of (13) are characterized by two numbers: the number of the state () of the localized part of the system and the number of the channel (), called environment, in which the system is embedded. Usually, and . Further, the wave functions and differ from one another due to the EM of the eigenstates via environments and , respectively. That means the system has four states, from a mathematical point of view.

The two environments are different from and orthogonal to one another. Further, the two eigenstates with equal and different arise from the same state of the localized part of the system. The zeros in the matrix (13) express the corresponding fact that the two eigenstates relative to the two channels 1 and 2 of cannot interact with one another.

We mention here that the Hamiltonian (13) is formally the same as the Hamiltonian (24) or rather in [10]. There is however a fundamental difference: in in [10], the transition of excitons is considered while (13) describes primarily the transition of particles. Furthermore, in in [10] the whole system is fully embedded into both environments () and () which are both of completely different nature and exist independently of one another. In contrast to this, the Hamiltonian (13) describes a system, the states of which are primarily embedded partially in each of the two different environments. These two different environments are also independent of one another. They are, however, nothing but parts of the total environment.

Without an EP in the considered parameter range in relation to both channels, we have , , and . Under the influence of an EP relative to and/or relative to , the eigenvalues and eigenfunctions will be, however, different from one another; , , and in the corresponding parameter range.

In analogy to (10), we will consider also the non-Hermitian Hamiltonian:which describes the system without any mixing of its states via any environment. In other words, corresponds to vanishing EM of the eigenstates via an environment. The mixing of the eigenstates of (13) can be represented in a set of eigenfunctions of (14) in complete analogy to the relation (11) for two states coupled to one common environment.

##### 2.4. Schrödinger Equation with

Using (14), we can write down the Schrödinger equation with source term for the two-channel case in analogy to (12) for the one-channel case. The corresponding equation readsThe source depends on the coupling of the system to both channels, i.e., on and on . It does not depend on the energies .

We repeat here that, according to their definition [1], EPs occur only in the one-channel case, i.e., only in the two submatrices related to channel and channel , respectively. They are not defined in the matrix (13). However, the EPs of the two submatrices in (13) may influence the dynamics of the open two-channel system.

##### 2.5. Non-Hermitian Hamiltonian and Resonance Structure of the Matrix

The resonance structure of the matrix containing two resonance states can be obtained from the expressionwhich is derived in [22] by rewriting the Breit-Wigner expression for one or more isolated resonances that are coupled to a common continuum of scattering wavefunctions; see also [7, 17]. The expression (16) is unitary. According to (16), the resonance structure of the cross section is determined exclusively by the spectroscopic values of the localized part of the system, i.e., by the eigenvalues of the non-Hermitian Hamiltonian . The expression (16) allows us therefore to receive reliable results also in the two-channel case and, moreover, when the phase rigidity of the eigenfunctions of is reduced, and when the eigenfunctions of contain EM, i.e., when they are mixed in the set of eigenfunctions of according to (11).

According to the results obtained in [17], the evolution of the system near to an EP is driven exclusively by the nonlinear source term of the Schrödinger equation (12) which describes the coupling of the localized part of the system to the common environment and is characteristic of the open quantum system embedded in one environment. The calculations in [17] are performed for fixed ; i.e., can not be responsible for the width bifurcation occurring in these calculations under the influence of an EP. The only conclusion is therefore that the nonlinear source term is the driving force. It is able, in the one-channel case, to largely conserve the resonance structure of the cross section.

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 [17]. This result agrees, on the one hand, with the experience received from many different numerical studies in realistic cases which are performed without taking into account EM. On the other hand, it is not in contradiction with the conclusions received from the study of non-Hermitian physics of open quantum systems.

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.

#### 3. Numerical Results

##### 3.1. Eigenvalues and Eigenfunctions of

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 . Knowing these results, we study then, in the following Section 3.2, the influence of the singularity onto observable values such as resonance structure and contour plot of the transmission probability.

In Figure 1, we provide numerical results for the parameter dependence of eigenvalues and eigenfunctions of the Hamiltonian . We consider the case that one of the states is coupled to a special channel much stronger than the other one, . Without loss of generality, we choose the extreme case ; ; .

We are interested, above all, in the appearance of a critical parameter value at which the spectroscopic values of the localized part of the system change. All our calculations show that is well defined: it is the value of maximum width bifurcation. At , the phase rigidity approaches the value , meaning that the two eigenstates of the non-Hermitian Hamiltonian become orthogonal at this parameter value. However, the EM of the states via the continuum cannot be neglected at .

In Figure 1 we show a few typical numerical results. The results of the left column are obtained with and those of the right column with . The other parameters are given in the figure caption.

The results are the following. The energies cross at (Figures 1(a) and 1(f)), while the widths bifurcate (Figures 1(b) and 1(g)). Notably, the state with vanishing width gets a positive width due to the width bifurcation; i.e., it does not decay around .

Around the critical parameter value, the phase rigidity decreases slightly (Figures 1(c), 1(d), 1(h), and 1(i)). It approaches the value 1 when . That means, the eigenfunctions of the non-Hermitian operator become orthogonal when . They are mixed via the environment (Figures 1(e) and 1(j)).

When we start from parameter independent energies and parameter dependent widths in contrast to the case considered in Figure 1, then the biorthogonal eigenfunctions of become orthogonal at maximum level repulsion. In any case, there is a critical parameter value at which the biorthogonal eigenfunctions of become orthogonal.

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.

##### 3.2. Resonance Structure and Contour Plot of the Transmission Probability

Observable information on the spectroscopic properties of the localized part of the system is contained in the resonance structure of the cross section. Using (16), we have calculated the resonance structure of the cross section at the critical parameter value and at the two values and of far from .

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 [7, 17, 23]. An example is shown in Figure 2. That means the cross section in the two-channel case is very much like that in the one-channel case when the system is coupled with the same strength to both channels and the EM of the states is the same. The results obtained for other values of the coupling strength (e.g., for ; or ) are very similar.

Additionally, we have performed some calculations with different values of the original widths () under the condition . As a result, differences in the values of the original widths almost do not influence the resonance structure of the cross section in the critical parameter range. This corresponds to the fact that are not involved in the source term of the Schrödinger equation (15) and therefore do not contribute to the EM of the states via the environment.

When the EM of the eigenstates via the continuum of scattering wavefunctions is different for the two channels and different from zero, , then the minimum of the cross section at will fill up. An example is shown in Figure 3. Similar results (not shown) are received with and . In both cases, several different values are used for . The filling-up of the minimum causes a broad maximum of the cross section (after averaging) around instead of the typical resonance peak. This can be seen well when .

Although the double-hump structure of the cross section, appearing under the condition , is well pronounced, it differs, even for small , from that obtained for vanishing coupling strength between system and environment (see Figure 4). This result differs from those obtained in the one-channel case; see [17]. That means, EM can generally not be neglected in the two-channel case.

Further information is contained in the contour plots of the cross section some of which are shown in the lower parts of Figures 2, 3, and 4. Most interesting is the padding of the minimum of the double-hump structure of the cross section in the critical parameter range around when the EM of the two states via the environment is different from one another. It causes an enhancement of the cross section around the critical parameter value .

In all cases, the two eigenstates of the non-Hermitian operator become orthogonal in approaching the critical value . Here the eigenfunctions of are mixed in the set of eigenfunctions of (14).

#### 4. Discussion of the Results and Conclusions

The results of our calculations show very clearly that not only the eigenvalues of play an important role in the non-Hermitian quantum physics of open quantum systems but also the eigenfunctions of are of similar importance. It depends on the physical problem that is considered, whether the eigenvalues are more significant than the eigenfunctions or vice versa. In any case, the eigenvalues* and* the eigenfunctions must be considered in describing and explaining a physical problem.

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 [17] and holds true also in the two-channel case as can be seen from the results in Figure 1.

Instead of an EP, we see in Figure 1 the critical parameter value at which the energy trajectories cross and the width bifurcation is maximum. Here, the phase rigidity approaches the value 1, meaning that the eigenfunctions are orthogonal. They are strongly mixed via the continuum. Similar results are very well known also from one-channel studies [17]. The meaning of this result is not fully understood up to now.

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 become orthogonal when width bifurcation (or level repulsion) becomes maximum.

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 at which the eigenfunctions of the non-Hermitian Hamilton operator become orthogonal. Other questions arise from the meaning of EPs in many-channel systems.

#### Conflicts of Interest

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

#### Acknowledgments

The authors thank Jon Bird for valuable discussions.

#### Endnotes

1. 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).2.In contrast to the definition that is used in, for example, nuclear physics, we define the complex energies before and after diagonalization of by and , respectively, with and for decaying states. This definition will be useful when discussing systems with gain (positive widths) and loss (negative widths).#### References

- T. Kato,
*Perturbation Theory for Linear Operators*, vol. 132 of*Grundlehren der Mathematischen Wissenschaften*, Springer, New York, NY, USA, 1966. View at: MathSciNet - C. Dembowski, H. Gräf, H. L. Harney et al., “Experimental Observation of the Topological Structure of Exceptional Points,”
*Physical Review Letters*, vol. 86, no. 5, pp. 787–790, 2001. View at: Publisher Site | Google Scholar - C. Dembowski, B. Dietz, H. Gräf et al., “Observation of a Chiral State in a Microwave Cavity,”
*Physical Review Letters*, vol. 90, no. 3, 2003. View at: Publisher Site | Google Scholar - T. Gao, E. Estrecho, K. Y. Bliokh et al., “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,”
*Nature*, vol. 526, no. 7574, pp. 554–558, 2015. View at: Publisher Site | Google Scholar - N. Moiseyev,
*Non-Hermitian Quantum Mechanics*, Cambridge University Press, Cambridge, UK, 2011. View at: Publisher Site | MathSciNet - N. Auerbach and V. Zelevinsky, “Super-radiant dynamics, doorways and resonances in nuclei and other open mesoscopic systems,”
*Reports on Progress in Physics*, vol. 74, Article ID 106301, 2011. View at: Google Scholar - I. Rotter, “A non-Hermitian HAMilton operator and the physics of open quantum systems,”
*Journal of Physics A: Mathematical and General*, vol. 42, no. 15, 153001, 51 pages, 2009. View at: Publisher Site | Google Scholar | MathSciNet - I. Rotter and J. P. Bird, “A review of progress in the physics of open quantum systems: theory and experiment,”
*Reports on Progress in Physics*, vol. 78, Article ID 114001, 2015. View at: Google Scholar - M. Müller and I. Rotter, “Phase lapses in open quantum systems and the non-Hermitian Hamilton operator,”
*Physical Review A*, vol. 80, Article ID 042705, 2009. View at: Google Scholar - H. Eleuch and I. Rotter, “Gain and loss in open quantum systems,”
*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*, vol. 95, no. 6, Article ID 062109, 11 pages, 2017. View at: Publisher Site | Google Scholar - W. D. Heiss, M. Müller, and I. Rotter, “Collectivity, phase transitions, and exceptional points in open quantum systems,”
*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*, vol. 58, no. 3, pp. 2894–2901, 1998. View at: Publisher Site | Google Scholar - O. Atabek, R. Lefebvre, M. Lepers, A. Jaouadi, O. Dulieu, and V. Kokoouline, “Proposal for a Laser Control of Vibrational Cooling in,”
*Physical Review Letters*, vol. 106, no. 17, 2011. View at: Publisher Site | Google Scholar - A. Jaouadi, M. Desouter-Lecomte, R. Lefebvre, and O. Atabek, “Signatures of exceptional points in the laser control of non-adiabatic vibrational transfer,”
*Journal of Physics B*, vol. 46, Article ID 145402, 2013. View at: Google Scholar - I. Gilary and N. Moiseyev, “Asymmetric effect of slowly varying chirped laser pulses on the adiabatic state exchange of a molecule,”
*Journal of Physics B*, vol. 45, Article ID 051002, 2012. View at: Google Scholar - G. A. Álvarez, E. P. Danieli, P. R. Levstein, and H. M. Pastawski, “Environmentally induced quantum dynamical phase transition in the spin swapping operation,”
*The Journal of Chemical Physics*, vol. 124, Article ID 194507, 2006. View at: Google Scholar - A. Ruderman, A. D. Dente, E. Santos, and H. M. Pastawski, “Molecular dissociation in presence of a catalyst: II. The bond breaking role of the transition from virtual to localized states,”
*Materials Research Express*, vol. 3, no. 8, Article ID 085017, 2016. View at: Google Scholar - H. Eleuch and I. Rotter, “Resonances in open quantum systems,”
*Physical Review A*, vol. 95, no. 6, Article ID 022117, 2017. View at: Publisher Site | Google Scholar - I. Rotter, “Dynamics of quantum systems,”
*Physical Review E*, vol. 64, no. 3, Article ID 036213, 2001. View at: Publisher Site | Google Scholar - B. Wahlstrand, I. I. Yakimenko, and K. Berggren, “Wave transport and statistical properties of an open non-Hermitian quantum dot with parity-time symmetry,”
*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*, vol. 89, no. 6, Article ID 062910, 2014. View at: Publisher Site | Google Scholar - F. Tellander and K. Berggren, “Spectra, current flow, and wave-function morphology in a model,”
*Physical Review A: Atomic, Molecular and Optical Physics*, vol. 95, no. 4, Article ID 042115, 2017. View at: Publisher Site | Google Scholar - J. Gros, U. Kuhl, O. Legrand, F. Mortessagne, E. Richalot, and D. Savin, “Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations,”
*Physical Review Letters*, vol. 113, no. 22, Article ID 224101, 2014. View at: Publisher Site | Google Scholar - I. Rotter, “Effective Hamiltonian and unitarity of the S matrix,”
*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*, vol. 68, no. 1, 016211, 13 pages, 2003. View at: Publisher Site | Google Scholar | MathSciNet - M. Müller, F. Dittes, W. Iskra, and I. Rotter, “Level repulsion in the complex plane,”
*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*, vol. 52, no. 6, pp. 5961–5973, 1995. View at: Publisher Site | Google Scholar

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Copyright © 2018 Hichem Eleuch and Ingrid Rotter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.