The physics of exceptional points
Abstract
A short resume is given about the nature of exceptional points (EPs) followed by discussions about their ubiquitous occurrence in a great variety of physical problems. EPs feature in classical as well as in quantum mechanical problems. They are associated with symmetry breaking for symmetric Hamiltonians, where a great number of experiments have been performed in particular in optics, and to an increasing extent in atomic and molecular physics. EPs are involved in quantum phase transition and quantum chaos, they produce dramatic effects in multichannel scattering, specific time dependence and more. In nuclear physics they are associated with instabilities and continuum problems. Being spectral singularities they also affect approximation schemes.
pacs:
03.65.Vf, 03.65.Xp, 31.15.p, 02.30.f, 02.30.Tb1 Introduction
Singularities of functions describing analytically observable quantities have always been at the scrutiny of theoretical investigations. For instance, the structures of measured cross sections are usually associated with pole terms in the complex energy plane of the scattering amplitudes. In turn, these pole terms are associated with specific boundary conditions of solutions of, say, the Schrödinger equation [1]. Another example is the pattern of spectra when plotted versus an external strength parameter, say, of a magnetic field; it usually shows the phenomenon of level repulsion, often associated with quantum chaos [2]. When such spectra are continued into the complex plane of the strength parameter, one encounters a different type of singularities where two repelling levels are connected by a square root branch point. If for real strength parameter the Hamiltonian is hermitian, the branch points always occur at complex parameter values thus rendering the continued Hamiltonian as nonhermitian. As a consequence, the wellknown properties associated with a degeneracy of hermitian operators are no longer valid. These singularities have been dubbed exceptional points (EPs) by Kato [3].
A few decades ago these singularities have been perceived as a mathematical phenomenon that would be ’in the way’ of approximation schemes [4, 5] (see also the discussion in Section 4 below). However, their physical significance has been recognised in an early paper by Berry [6] based on an observation by Pancheratnam [7]. The specific algebraic property of the dielectric tensor (it cannot be diagonalised) brings about particular physical effects. This has been expounded in great detail in [8] for particular optical systems. As is discussed below optical systems constitute one major realm where EPs have a major bearing. Yet, originating from a parameter dependent eigenvalue problem EPs naturally occur and can give rise to dramatic effects in a great variety of physical problems: in mechanics, electromagnetism, atomic and molecular physics, quantum phase transitions, quantum chaos and more.
In Section 3 we return to these phenomena in detail. In 3.1 we discuss the experimental manifestation using microwave cavities where all the mathematical properties have been confirmed to be a physical reality. The investigation and physical realisation of symmetric Hamiltonians has become a major subject of theoretical and experimental endeavor. An important role is played by EPs in this research, it is dealt with in 3.2. Subsection 3.3 is devoted to a short discussion of the role of EPs in atomic and molecular physics while in 3.4 recent findings in open systems and laser physics are mentioned. In 3.5 we demonstrate the essential role of EPs in understanding the dramatic effects associated with a quantum phase transition and quantum chaos, while 3.6 presents a short discussion of multichannel scattering and 3.7 touches upon EPs of higher order. Classical mechanics and fluids are briefly dealt with in 3.8. The sometimes adverse role of EPs in approximation schemes is, for historical reasons, discussed in Section 4.
For the sake of completeness we rehash the essential formal background in the following Section 2. The last Section 5 gives a summary.
2 Exceptional points
Exceptional points occur generically in eigenvalue problems that depend on a parameter. By variation of such parameter (usually into the complex plane) one can generically find points where eigenvalues coincide.
In the immediate vicinity of an EP the special algebraic behaviour – as discussed below – allows a reduction of the full problem to the two dimensional problem associated with the two coinciding levels [9]. We thus confine our discussion to the eigenvalues of a twodimensional matrix where the direct connection of an EP and the phenomenon of level repulsion is easily demonstrated. Consider the problem
(1)  
where the parameters and determine the noninteracting resonance energies . We may choose all parameters complex and we require to ensure the problem to be nontrivial. Owing to the interaction invoked by the matrix elements the two levels do not cross but repel each other. However, the two levels coalesce at specific values of in the vicinity of the level repulsion, that is at the two EPs
(2)  
(3) 
For the energy levels have a square root singularity as a function of and read
(4) 
and the eigenvalues at the EPs are
(5) 
We use the term coalesce as the pattern is distinctly different from a degeneracy usually (but not only) encountered for hermitian operators. Note that is hermitian only when , and are real and . At the EP the difference between a degeneracy and a coalescence is clearly manifested by the occurrence of only one eigenvector instead of the familiar two in the case of a genuine degeneracy. Using the biorthogonal system for a nonhermitian matrix, the (only one) right hand eigenvector reads at (up to a factor)
(6) 
and at
(7) 
while the corresponding left hand eigenvector is at and
(8)  
(9) 
respectively. Note that the norm – that is the scalar product – vanishes which is often referred to as selforthogonality [10]. It is this vanishing of the norm that enables the reduction of a high dimensional problem to two dimensions in close vicinity of an EP [11]. The existence of only one eigenvector with vanishing norm is related to the fact that for or the matrix cannot be diagonalised [3]. At these points the Jordan decomposition reads
(10) 
with
(11) 
Similar expressions hold at . We mention that the second column of is often referred to as an associate vector obeying .
If one or both vanish there is level crossing (). If only or vanishes then and thus cannot be diagonalised at the crossing point, therefore the Jacobian form is nondiagonal and only one eigenvector exists at the crossing point; yet it is not an EP as there is no square root singularity in . We do not further pursue such case as it appears to be of no physical interest.
The square root singularity also affects the the Green’s function and thus the scattering matrix in a particular way. At the EPs the Green’s function (and scattering matrix) have a pole of second order [12] in addition to the familiar pole of first order. For the case considered here the explicit form of the Green’s function reads at
(12) 
and similar at . The first term resembles the conventional expression at a nonsingular point of the spectrum; the second term is a consequence of the singular spectral point. This term can give rise to dramatic effects near to the EP [13], and at the EP special effects are generated from the interference of the two terms when the square of the modulus of the scattering matrix is considered.
The square root behaviour as given by Eq.(4) has further physical consequences as discussed in the following sections. Here we list the major mathematical reasons giving rise to special physical phenomena.

In the vicinity of an EP the spectrum is strongly dependent on the interaction parameter; in fact, the derivative with respect to of the eigenvalues and eigenvectors is infinity at the EP.

Encircling or in the plane will interchange the two energy levels (see Fig.1).

Avoiding the EP there are two linearly independent eigenvectors and (and the left hand companions and ) with the normalisation . Enforcing the normalisation into the EP (where vanishes) the components of the eigenvectors tend to infinity as . Repeated encircling – say counterclockwise – of an EP generates the pattern for the normalised eigenvectors
(13) from which the pattern for follows accordingly [9, 14]; the fourth root behaviour is clearly seen as four rounds are needed to reach the initial sheet. Notice that these relations imply a kind of chiral behaviour: going clockwise instead of counterclockwise gives a different sign. We note that these relations are presented inconsistently in [15].

When the eigenfunctions and coalesce as given in (6) and (7) (here we normalise the vectors by setting one component equal to unity), they become, for , independent of parameters and assume the form
The phase difference of between the two components is changed if, for instance, time reversal symmetry is broken by choosing complex [16, 17]. Note further that must be complex for hermitian operators and in which case the two EPs occur at complex conjugate values, i.e. . In general or can be real [18]. Recall that an EP can be approached in the laboratory only if the corresponding energy has a nonpositive imaginary part.
3 Physical effects
Many cases of particular effects have been reported in the literature during the past ten years. We here discuss only some major trends and developments. While we focus the discussion upon quantum mechanical problems and optics, the ubiquitous character of EPs in any parameter dependent eigenvalue problem makes them appear also in problems of classical mechanics and others.
3.1 Microwave cavity
Probably for the first time ever the direct encircling of the square root branch point  that is the manifestation of the two Riemann sheets (see Fig.1)  was accomplished with a microwave resonator [19].
The realisation of the complex parameter was implemented in the laboratory by two real parameters: (i) the coupling between the two halves of the cavity and (ii) the variation of the one level in one half of the cavity. In one experiment the direct approach of the EP was avoided while the encircling was done at close distance. All properties as listed in the previous section have been confirmed: encircling an EP swaps the energies and fourfold encirclements yield relations (13).
The chiral property of the wave function at the EP has been confirmed in a second experiment [20] where the phase difference of between the two components has been measured in a direct approach of an EP.
The same results have been established with two coupled electronic circuits [21].
More recent experiments implementing carefully time reversal symmetry breaking by using a magnetised device at the coupling of two cavities [16] are in perfect accordance with the predictions made in [17].
Other types of experiments and/or theoretical investigations with microwave cavities are found in [22, 23] where the effects of EPs feature prominently. The study of continuous variation of the coupling between a cavity and a rubidium atom enabled the direct observation of an EP in an open quantum system [24].
3.2 symmetric Hamiltonians
It has been suggested to extent the class of the traditional hermitian Hamiltonians by a specific choice of nonhermitian operators [73]. Hamiltonians that are symmetric under the combined operation of parity and time reversal transformation (symmetric operators) can have a real spectrum even though the operators can be nonhermitian. If the eigenstates are also symmetric under , that is if , the eigenvalues are real; when the symmetry is broken, that is if the above relation does not hold, the eigenvalues are complex [26]. It turns out that the parameter values where the symmetry breaking occurs are just those values where an EP of the system appears. Since the Hamiltonians considered in this class can be nonhermitian, the EPs can occur for real parameter values. Obviously, this particular extension of traditional quantum mechanics has attracted great interest in the literature with umpteen theoretical and experimental papers.
Before turning to a selection of these we rehash some basic features of the specific class of operators considered here. While unbounded nonHermitian operators constitute a difficult mathematical problem in general, we here deal with the much simpler class that is well understood: the quasiHermitian operators [27]. A nonHermitian operator is called quasiHermitian if there exists a bounded Hermitian positivedefinite operator that ensures the relation
(14) 
The relation implies that is similarly equivalent to a Hermitian operator, in fact the operator
(15) 
is Hermitian when is the positive root of . The operator may be viewed as a metric characterising a different Hilbert space by defining the scalar product
(16) 
where is the usual scalar product, employing the metric being the identity. Two important observations can be made: (i) the nonhermitian operator is in fact hermitian if the new metric is used for the underlying Hilbert space; note, however, that then other operators (like position or momentum) may no longer be necessarily hermitian. There is freedom though in the choice of the metric [27, 28]; (ii) in view of (15) the spectrum of is real.
Of particular interest in our context is the fact that at a parameter value where the symmetry gets broken – at the EP –, the singularity affects the the metric as well; in fact the metric ceases to exist [29]. While the occurrence of an EP has been shown to cause a great variety of physical effects, the break down of the metric has been associated with a speculative suggestion: that the transition time between the two states connected at the EP may be much shorter than the expected time [30].
Many theoretical papers have been published on the subject dealing with the mathematical aspects of diagonalisable nonhermitian operators [31]  [34]; the list given here can only be incomplete while the quoted papers contain further pertinent references. More recent theoretical papers deal with specific subjects such as symmetry in optics [35] and nonlinear wave equations [36]. In [37] a new class of chaotic systems with dynamical localization is studied: a gain/loss parameter invokes a spontaneous phase transition from real values of the spectrum (phase of conserved symmetry) to complex values (phase of broken symmetry).
Perhaps some of the most beautiful demonstrations of symmetry breaking at an EP have been made in optical systems. Light propagation is used in distributedfeedback optical structures with gain or loss regions. The EPs at the points of symmetry breaking of the Dirac Hamiltonian give rise to simple observable physical quantities such as resonance narrowing and laser oscillation [38]. In symmetric optical lattices the transition at the spectral singularity (EP) has been demonstrated [39] (see also [40][44]). More experimental work along these lines can be expected. In particular specific quantum mechanical systems like the ones discussed in the following subsection are expected to involve aspects of symmetry.
An interesting twist is the classical analogue to quantum mechanical symmetric Hamiltonians [45]. It appears that the correspondence principle can be usefully extended into the complex domain of classical variables. The association between complex quantum mechanics and complex classical mechanics is subtle and requires a nonperturbative approach (see also [29, 46]).
3.3 EPs in atomic/molecular physics, Feshbach resonances
Using Feshbach resonance techniques there are recent proposals for resonant dissociation by lasers of molecules or alkali dimers, where the effects of EPs are expected to feature prominently [49, 50]. Similar in spirit, a BoseEinstein condensate of neutral atoms with induced electromagnetic attractive () interaction has been discussed recently as another system allowing a tunable interaction [51]. The critical value  an EP  where the onset of the collapse of the condensate occurs is interpreted as a transition point from separate atoms to the formation of molecules or clusters [52]. In this context we recall that level splitting by potential barriers (quantum tunneling) is associated with a coalescence of the two levels at an appropriate (complex) value of the barrier strength.
In a recent investigation it was found that by transporting an eigenstate around an EP the final eigenstate may be different from the initial eigenstate. The precise interplay between gain/loss and nonadiabatic couplings imposes specific limitations on the observability of this flip or nonflip effect [53].
3.4 EPs in laser physics and open systems
EPs can strongly affect the abovethreshold behavior of laser systems. They are induced by pumping the laser nonuniformly [54]. In the vicinity of the EPs, where the laser modes coalesce, the effect of the singularities can explain the turning off of one laser even when the overall pump power deposited in the system is increased.
The relation between EPs in closed and open gainloss structures has been expounded recently [55]. The EPs occurring for a closedsystem Hamiltonian are related to those of the scattering matrix for the corresponding open system. It has been demonstrated in detail how these seemingly different situations share a very close and elegant connection with each other.
3.5 Quantum phase transitions, chaos
The important role played by EPs in connection with quantum phase transition and also quantum chaos in many body systems appears to be lesser appreciated by the ’EPcommunity’. We here report some pertinent results. The Lipkin model [56] is a toy model often used to study quantum phase transitions of many body systems. The interaction of the two level model lifts or lowers a Fermion pair between the two levels. For particles it can be formulated in terms of the angular momentum operators and reads
(17) 
with being the dimensional representations of the operators. There is a phase transition at that moves toward in the thermodynamic limit (). The special interaction gives rise to an inherent symmetry: even and odd numbers of the ordered levels do not interact. The phase for is the ’normal’ phase where the symmetry of the problem is preserved by the levels and wave functions. In the ’deformed’ phase for the symmetry is broken in that even and odd become degenerate. Here the role of the EPs is crucial to bring about the phase transition in the spectrum [57, 58]. In Fig.2 the pattern of the EPs is illustrated for low values of . It is clearly seen how the EPs accumulate for increasing on the real axis with the tendency to move towards the point . The spectrum remains unaffected by singularities in the region of the normal phase while it is strongly affected around the critical point. For finite temperature these singularities feature in the partition function as is discussed explicitly in [59]. If the model is perturbed the regular pattern of the EPs is destroyed and so is the spectrum accordingly. The onset of chaos [60] is clearly discernible in the region of the phase transition associated with a high density of EPs while the model remains robust outside the critical region for sufficiently mild perturbation.
3.6 Special effects in multichannel scattering
Depending on a judicious choice of parameters the proximity of EPs can invoke dramatic effects in multichannel scattering such as a sudden increase of the cross section in one channel, even by orders of magnitude. In turn, a second channel is suppressed and can show a resonance curve that deviates substantially from the usual Lorentz shape [13]. Related to this behaviour is the pattern in the time domain [61]. Depending on the initial conditions the wave function displays characteristic features such as very fast decay or the opposite, i.e. very long life time. At the EP the time dependent wave function typically has a linear term in time besides the usual exponential behaviour.
3.7 EPs of higher order
The coalescence of three levels [62] (or more) is of course possible if sufficiently many parameters are at one’s disposal. For an EPN ( levels coalescing) an dimensional matrix must be considered. For complex symmetric matrices parameters are needed to enforce the coalescence of levels [63]. An interesting aspect is the behaviour of, say, an EP3 where two EP2 sprout out from the EP3 under variation of one of the parameters; in this way we can view an EP3 as the coalescence of two EP2 being linked by their position on one common Riemann sheet. This can be generalised allowing many combinations for larger : the block structure of the Jordan form of the Hamiltonian gives an indication of the connectedness of the levels. In a recent paper [64] this is investigated for EP3; a physical example for possible implementation in the laboratory is suggested. Topological properties and their group structure of higher order EPs are investigated in [65].
3.8 Classical systems
Even though some of the experiments with microwave cavities as well as some of the optical systems are classical in nature we here note in particular effects of EPs in classical [66] and fluid mechanics. Instabilities and a particular behaviour of the Reynold number for a Poiseuille flow have been associated with near crossings of eigenvalues and – as is now identified – with EPs [67, 68] (see also [69]).
4 The role of EPs in approximation schemes
The well known Random Phase Approximation (RPA) used in many body problems yields an effective Hamiltonian that is nonhermitian [70]. As a result, eigenvalues are not necessarily real. Depending on the strength of the, say, particlehole interaction two real eigenvalues and coalesce at and then move into the complex plane when the interaction is increased. Often this instability point is associated with one or more phase transition of the underlying mean field [71]. It is an EP with all its characteristics: square root branch point in the interaction strength and the vanishing norm of the wave function.
A perturbative approach in shell model calculations can be hampered by singularities associated with intruder states [72]. These singularities are EPs where two levels coalesce thus limiting the radius of convergence of the perturbation series. In a similar vein, branch point singularities (EPs) have been identified to affect the convergence of perturbation series in a field theoretical model [73] and for the anharmonic oscillator [74]; in these infinite dimensional cases accumulation points of EPs are encountered.
Recent approaches to model nuclei near to the drip line [75] use resonance states to describe the continuum. The coalescence of two resonances can invoke specific physical effects owing to the strong increase of the associated spectroscopic factors being caused by the vanishing norm of the wave functions at the EP.
5 Summary
The ubiquitous occurrence of EPs in all eigenvalue problems that depend on a parameter can have significant and often dramatic effects on observables in a great variety of physical phenomena. A few decades ago, these singularities appeared as a purely mathematical feature that could cause problems in approximation schemes. It was only about fifteen years ago that their physical manifestation has been demonstrated in experiments that were basically classical in nature. At present definite theoretical and experimental proposals are found in the literature relating to atomic and molecular physics, using lasers for triggering and measuring specific transitions. In nuclear physics, where there is now great interest in open systems, that is in nuclei on the drip line, the coalescence of resonance states is expected to produce specific effects such as enhancements of particular reactions. The tremendous upbeat of research in connection with symmetric Hamiltonians is expected to further proliferate new effects and applications of EPs also and in particular in genuine quantum mechanical problems.
References
References
 [1] R.G. Newton 1967 Scattering of Wave and Particles, McGraw Hill
 [2] see for instance D. Wintgen and H. Friedrich 1986, Phys. Rev. Lett. 57 571
 [3] T. Kato 1966, Perturbation theory of linear operators Springer, Berlin
 [4] W.D. Heiss 1970, Nucl.Phys. A144, 417
 [5] N. Moiseyev and S. Friedland 1980, Phys.Rev. A 22, 618
 [6] M.V. Berry 1994, Current Science 67, 220
 [7] S Pancharatnam 1955, Proc.Ind.Acad.Sci. XLII, 86
 [8] M.V. Berry and D.H.J. O’Dell 1998, J.Phys.A33, 2093
 [9] W.D. Heiss 1999, Eur.Phys.J. D 7 1 ; W.D. Heiss 2000, Phys.Rev. E 61 929
 [10] N. Moiseyev 2011 NonHermitian Quantum Mechanics, Cambridge: Cambridge University Press
 [11] W.D. Heiss and H.L. Harney 2001, Eur. Phys. J. D 17 149
 [12] E. Hernandez, A. Jaureguit and A. Mondragon 2000, J. Phys. A 33, 4507
 [13] W.D. Heiss and R. G. Nazmitdinov 2010, Eur.Phys.J. D58, 53
 [14] H. MehriDehnavi and A. Mostafazadeh 2008, J. Math. Phys., 49, 082105
 [15] I. Rotter 2009, J. Phys. A: Math. Theor. 42, 153001
 [16] B. Dietz, H.L. Harney, O.N. Kirillov, M. MiskiOglu, A. Richter and F. Schäfer 2011, Phys.Rev.Lett. 106, 150403
 [17] HL. Harney and W. D. Heiss 2004, Eur.Phys.J. D 29, 429
 [18] W.D. Heiss 2004, J. Phys. A: Math. Theor. 37, 2455
 [19] C. Dembowski, H.D. Gräf, H.L.Harney, A.Heine, W.D. Heiss, H. Rehfeld and A.Richter 2001, Phys.Rev.Lett. 86, 787
 [20] C. Dembowski, B. Dietz, H.D. Gräf, H. L. Harney, A. Heine, W.D. Heiss and A. Richter 2003, Phys.Rev.Lett. 90, 034101; 2004 Phys.Rev. E 69, 056216
 [21] T. Stehmann, W.D. Heiss and F.G. Scholtz 2004, J.Phys. A: Mathematical & General, 37, 7813
 [22] SangBum Lee etal. 2009, Phys.Rev.Lett 103, 134101
 [23] J. Wiersig, S.W. Kim and M. Hentschel 2008, Phys. Rev. A 78 053809, J. Wiersig, 2011, Phys.Rev. A 84, 063828
 [24] Youngwoon Choi etal. 2010, Phys.Rev.Lett 104, 153601
 [25] C.M. Bender and S. Boettcher 1998, Phys.Rev.Lett. 80, 5243
 [26] P. Dorey, C. Dunning and R. Tateo 2001, J.Phys. A: Math. Gen. 34 L391
 [27] F. G. Scholtz, H.B. Geyer and F. J. W. Hahne, Ann. Phys. (N.Y.) 213, 74 (1992)
 [28] H.B. Geyer, W.D. Heiss and F.G. Scholtz 2008, Can.J.Phys. 86, 1195
 [29] D. P. Musumbu, H. B. Geyer and W. D. Heiss, J. Phys. A: Math. Theor. 40, F75 (2007)
 [30] C. M. Bender, D.C. Brody, H.F. Jones and B.K. Meister, Phys. Rev. Lett. 98, 040403 (2007)
 [31] A. Mostafazadeh 2002, J. Math. Phys. 43, 205 ; ibid 43, 2814; ibid 43, 3944
 [32] C. M. Bender 2007, Rep. Prog. Phys. 70, 947
 [33] M. Znojil 2006, J.Phys. A: Math. Gen. 39, 441
 [34] U. Günter, I. Rotter, and B.F. Samsonov 2007, J. Phys. A 40, 8815
 [35] EM Graefe and H.F. Jones 2011, Phys. Rev. A 84, 013818
 [36] B. Bagchi and A. Fring 2008, J. Phys. A: Math. Theor.41, 392004
 [37] C. West, T. Kottos, T. Prosen 2010, Phys.Rev.Lett. 104, 054102
 [38] S. Longhi 2010, Phys.Rev.Lett. 105, 13903
 [39] K.G. Makris, R. ElGanainy, D.N. Christodoulides and Z.H. Musslimani 2010, Phys.Rev. A 81, 63807
 [40] A.Guo etal. 2009, Phys.Rev.Lett. 103, 093902
 [41] Z. Lin etal. 2011, Phys.Rev.Lett. 106, 213901
 [42] Y. D. Chong, L. Ge, and A. D. Stone 2011, Phys.Rev.Lett. 106, 093902
 [43] A. Mostafazadeh 2009, Phys.Rev.Lett. 102, 220402
 [44] O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro 2009, Phys.Rev.Lett. 103, 030402
 [45] C.M.Bender, D. Hook, P. Meisinger, Q. Wang 2010, Phys. Rev. Lett. 104, 061601
 [46] A. Mostafazadeh 2006, J. Phys. A: Math. Gen. 39 10171
 [47] J. Schindler, A. Li, M.C. Zheng, F.M. Ellis and T. Kottos 2011, Phys. Rev. A 84, 040101
 [48] O.N. Kirillov 2012, Phys.Lett. A 376, 1244; O.N. Kirillov and F. Verhulst 2010, Z. angew. Math. Mech. 90, 462
 [49] R. Lefebvre and O. Atabek 2010, Eur.Phys.J. D 56 317; O. Atabek, R. Lefebvre, M. Lepers, A. Jaouadi, O. Dulieu and V. Kokoouline 2011, Phy.Rev.Lett. 106 173002
 [50] R. Lefebvre, O. Atabek, M. Sindelka and N. Moiseyev 2009, Phys.Rev.Lett 103, 123003
 [51] I. Papadopoulos, P. Wagner, G. Wunner, and J. Main 2007, Phys.Rev. A 76, 053604
 [52] H. Cartarius, J. Main, and G. Wunner 2008, Phys.Rev. A 77, 013618
 [53] R. Uzdin,A. Mailybaev and N. Moiseyev 2011, J. Phys. A: Math. Theor. 44, 435302
 [54] M. Liertzer, Li Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter 2012, Phys.Rev.Lett. 108, 173901
 [55] S. Rotter 2012, Exceptional points in open and closed gainloss structures, presented at the PHHQP11conference at Paris, 2731 August, to be submitted for publication
 [56] H.J. Lipkin, N. Meshkov and N. Glick 1965, Nucl.Phys. A 62, 188 ; ibid. 62, 199 (1965), ibid. 62, 211 (1965)
 [57] W.D. Heiss, F.G. Scholtz and H.B. Geyer 2005, J.Phys. A: Mathematical & General, 38, 1843
 [58] F. Leyvraz and W.D. Heiss 2005, Phys.Rev.Lett. 95, 050402
 [59] P. Cejnar, S. Heinze, M. Macek 2007, Phys.Rev.Lett. 99, 100601
 [60] W.D. Heiss and A.L. Sannino 1991, Phys.Rev. A 43, 4159
 [61] W.D. Heiss 2010, Eur.Phys.J. D60, 257
 [62] A.A. Andrianov, F. Cannata and A.V. Sokolov 2007, Nucl. Phys. B 773, 107
 [63] W.D. Heiss 2008, J. Phys. A: Math. Theor. 41, 244010
 [64] G. Demange and E.M. Graefe 2012, J. Phys. A: Math. Theor. 45, 025303
 [65] JungWan Ryu, SooYoung Lee, and Sang Wook Kim 2011, arXiv:1109.4216v1 [quantph]
 [66] A.P. Seyranian and A.A. Mailybaev 2003, Multiparameter stability theory with mechanical applications, Singapore: World Scientific
 [67] C.A. Jones 1988, Quart. J. Mech. Appl. Math. 41, 363
 [68] A.C. Or 1991, Quart. J. Mech. Appl. Math. 44, 559
 [69] F.H. Busse and A.C. Or 1986, J. Fluid Mech. 166, 173
 [70] P. Ring and P. Schuck 1980, The Nuclear Many Body Problem, Springer, New York
 [71] W.D. Heiss and R.G. Nazmitdinov 2002, Phys.Rev. C 65, 054304, and papers quoted therein
 [72] T.H. Schucan and H.A. Weidenmüller 1972 Ann.Phys.(NY) 73 108
 [73] C. M. Bender and T. T. Wu 1968 Phys. Rev. Lett. 21, 406
 [74] C. M. Bender and T. T. Wu 1969 Phys. Rev. 184, 1231
 [75] N. Michel, W. Nazarewicz, M. Ploszajczak and T. Vertse 2009, J. Phys. G: Nucl. Part. Phys. 36, 13101; N. Michel, W. Nazarewicz, J. Okolowicz, and M. Ploszajczak 2010, ibid. 37, 064042