AP07_2-3.vp 1 Introduction Non-Hermitian Hamiltonians with a complex eigenvalue spectrum have been studied almost since the formulation of quantum mechanics, most prominently as consistent de- scriptions of dissipative systems resulting for instance from channel coupling [1]. It has also been known for a very long time that many interesting non-Hermitian Hamiltonians with a real eigenvalue spectrum result naturally in various circum- stances. For instance, it was argued more than thirty years ago that the lattice versions of Reggeon field theory [2] � � � � � � � �a a iga a a a g a a a i i i i i i i j i � � � � � � � � � � † † † † † ( ) ~ ( )( i j ij i a � � � � � � � � � � � �� � ) (1) with a a i i � � †, being standard creation and annihilation ope- rators and �, , ~g g � �, possess a real eigenvalue spectrum [3] (I am grateful to John Cardy for pointing this out). The re- duction of the Hamiltonian in (1) to a single lattice site in zero transverse dimension [4] is very reminiscent of the so-called Swanson model [5], which results by replacing the interaction term with a simpler bilinear expression ga a g aa† † ~� . The lat- ter model serves currently as a concrete popular solvable model to exemplify various general features related to the study of non-Hermitian Hamiltonians [5, 6, 7, 8, 9]. Affine Toda field theory with a complex coupling constant is a very prominent class of field theoretical models, which are argued [10, 11] to be consistent despite their Hamiltonians being non-Hermitian. Besides the study of such explicit models re- lated to non-Hermitian Hamiltonians, in particular their spectral properties, the question of how to formulate the cor- responding quantum mechanical description consistently was first addressed in [12]. A useful insight into how to implement ��-symmetry into this formulation was later obtained [13]. The current large interest in the subject of non-Hermitian Hamiltonian systems was initiated about nine years ago [14] by the surprising numerical observation that even the class of simple non-Hermitian Hamiltonians � � �p g iz N2 ( ) , (2) defined on a suitable domain, possesses a real positive and discrete eigenvalue spectrum for integers N 2 with g � �. Supported by numerous new results and insights (for some recent reviews see [15, 16, 17, 18]), which have been obtained since, the natural question arises of how to construct non- -Hermitan Hamiltonians with real eigenvalue spectra in a more systematical way. The question I would like to address in this paper is how this may be achieved, in particular by generalizing some integrable models. 2 Real spectra of non-Hermitian Hamiltonians The activities in spectral theory usually focus on normal or self-adjoined operators in some Hilbert space. With regard to the remarks made in the introduction we shall first briefly re- view some arguments which may be used to explain the reality of the spectra of non-Hermitian Hamiltonians and then em- ploy them to construct new models, which depending on the argument used are guaranteed, or at least are likely, to have a real eigenvalue spectrum. 2.1 Pseudo-Hermiticity Since a Hermitian operator, say h h� †, is guaranteed to have real eigenvalues, i.e. h� ��� with � � �, one may trivially construct isospectral Hamiltonians by means of a similarity transformation H h� �� �1 , such that H� �� � with � � �� �1 . When � is a Hermitian operator this implies that the conjuga- tion of H is simply achieved by H H† � �� �2 2. Hamiltonians of this type are denoted as pseudo Hermitian Hamiltonians [12, 19, 20, 21, 22]. One of the immediate virtues of the these rela- tions is that �2 can be used consistently as a metric operator. Given a Hermitian Hamiltonian it is of course trivial to construct several isospectral non-Hermitian Hamiltonians in this manner simply by computing � �� �1h H for some posi- tive �. However, interesting situations arise when given simple non-Hermitian Hamiltonians, such as (1) and (2), possible to- gether with the knowledge that they possess a positive real 44 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 ��-symmetry and Integrability A. Fring We briefly explain some simple arguments based on pseudo Hermiticity, supersymmetry and ��-symmetry which explain the reality of the spectrum of some non-Hermitian Hamiltonians. Subsequently we employ ��-symmetry as a guiding principle to construct deformations of some integrable systems, the Calogero-Moser-Sutherland model and the Korteweg deVries equation. Some properties of these models are discussed. Keywords: ��-symmetry, pseudo-Hermiticity, Calogero-Moser-Sutherland model, KdV-equation. spectrum, and one tries to construct their Hermitian counter- parts by seeking convenient Hermitian operators �, such that � �H h h� � �1 †. Unfortunately, this is only feasible in an ex- act manner in some very rare cases [23, 5, 6, 7, 8, 9] and mostly one has to rely on perturbation theory, see e.g. [24, 25, 26, 8, 27]. More awkward is the fact that when given exclu- sively the non-Hermitian Hamiltonian H, there might be sev- eral Hermitian Hamiltonians counterparts and the metric is therefore not even uniquely determined. One may select out a particular metric by specifying for instance at least one more observable [12] or the spectrum. 2.2 Supersymmetry Another standard procedure produces isospectral Hamil- tonians is to employ Darboux transformations or equivalently a supersymmetric quantum mechanical construction [28, 29]. For this one considers Hamiltonians �, which can be decom- posed into the form � � � � �� �H H QQ Q Q ~ ~ (3) As indicated in (3) one assumes that the two superpartner Hamiltonians H� factor into the two supercharges Q and ~ Q , which intertwine the Hamiltonians H� as QH H Q� �� and ~ ~ Q H H Q� �� . Evidently the two charges commute with the Hamiltonian �, i.e. [ , ] [ , ~ ]� �Q Q� � 0 , and thus the sl(1/1) algebra constitutes a symmetry of �. As pointed out by vari- ous authors [30, 31, 32, 33, 34, 35, 36], one does not require the Hamiltonians H� to be Hermitian, such that we allow H H� �� † . The only constraints which are natural to impose when one wishes to make contact with the pseudo-Hermitian treatment in the previous section are that the individual fac- tors of H� are conjugated as [34] Q Q† ~ � � �� � �2 2 and ~ ~†Q Q� � �� � �2 2, (4) where the operators �� are Hermitian � �� �� † . As an im- mediate consequence of (4), both Hamiltonians H� in (3) become pseudo-Hermitian and possess Hermitian counter- parts h h� �� † H H h H� � � � � � � �� � � † � � � � � �2 2 1. (5) By construction all four Hamiltonians h�, H� are therefore isospectral H� � ��� �� and h� � ��� � � (6) and their corresponding wavefunctions are intimately related � � � � � � �� �Q � � 1 and � �� � � � �� � ~ Q � �1 . (7) One may now characterize four qualitatively different cases depending on the properties of the Hermitian opera- tors �� in (7), namely i) for generic �� we have isospectral quartets, ii) for generic �� and �� � � and iii) for generic �� and �� � � we find isospectral triplets and finally iv) for �� � � we have isospectral doublets. The interesting cases ii) and iii), which contain a Hermitian Hamiltonian, have been considered in [31]. Next, one needs to specify the explicit representation for the supercharges in terms of the superpotential W(x). Setting the parameter �2 2 1m � , the simplest choices are differential operators of the first order Q x W� � d d and ~ Q x W� � � d d (8) such that the two superpartner Hamiltonians may be written as H W W V� �� � � � � � � �� � 2 . (9) Alternative choices with higher order differential opera- tors are discussed for instance in [37]. Assuming further that H� possesses a discrete spectrum H n n n� � ��� �� , one may adjust the energy scale such that H n� � �� 0 for some chosen m. In order to single out this groundstate wavefunction we de- note it as m m mc W x: exp� � � � �� � � �� d , c � �. Consequently the superpartner potentials may be expressed in terms of the groundstate wavefunctions and acquire the forms Wm m m � � � , V m m m � � � �� , V m m m m m � � �� � �� � � �� � �� 2 2 . (10) Therefore the Hamiltonians H V E W W Em m m m m m� �� � � � � � � � � �� � 2 (11) are isospectral H Em n n n� � ��� � for n m� . (12) In order to disentangle the Hermitian from the non- -Hermitian case, we separate the superpotential into its real and imaginary part W w iwm m m� � � with w wm m� † , � � †w wm m� and likewise for the groundstate energy E im m m� �� �� . With these notations we can re-write (11) as H w w w i w w w m m m m m m m m m � � � � � � � � � � � � � � ( � � � ). � �2 (13) Clearly we encounter the situation ii) or iii) when w w wm m m m � � � �� � � � 2 or �wm � 0 , (14) respectively. When given a Hamiltonian, irrespective of being Hermi- tian or non-Hermitian, and at least one wavefunction, the exploitation of supersymmetry is a very constructive proce- dure to obtain isospectral Hamiltonians, which could also be Hermitian or non-Hermitian. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 45 Acta Polytechnica Vol. 47 No. 2–3/2007 2.3 ��-symmetry A further very simple and transparent way to explain the reality of the spectrum of some non-Hermitian Hamiltonians results when we encounter unbroken ��-symmetry, which in the recent context was first pointed out in [13]. This means that both the Hamiltonian and the wavefunction re- main invariant under a simultaneous parity transformation � : x x� � and time reversal � : t t� � , that is we require � �H, �� � 0 and �� � �� , (15) where � is a square integrable eigenfunction on some do- main of H. It is crucial to note that the ��-operator is an anti-linear operator, i.e. it acts as �� �� ��( ) * * � �� � � �� � � with �, � � and �, � being some eigenfunctions. An easy way to convince oneself of this property is to consider the standard canonical commutation relation [ , ]x p i� . Since �� : ,x x p p� � � , we require �� : i i� � : to keep this relation invariant. Utilizing now both relations in (15) and the anti-linear nature of the ��-operator, a very simple argument leads to the reality of the spectrum � � � �� � � � � � �� � � � � �H H H�� �� �� ��* * (16) Whereas the first relation in (15) is usually trivial to check, the second is in general difficult to access as one rarely knows all the wavefunctions. In case it does not hold, one speaks of a broken ��-symmetry and the eigenvalues come in complex conjugate pairs. All arguments in this subsection were essen- tially already known to Wigner in 1960 [38] relating to anti- -linear operators in a completely generic form. Noting that the ��-operator is an example of such an operator these ideas have been revitalized in a modified form and developed fur- ther in the recent context of the study of non-Hermitian Hamiltonians [13]. 3 ��-symmetry as a guiding principle to construct new models If we now wish to construct new models with real eigen- value spectra, we may in principle use any of the previous arguments. Clearly the exploitation of ��-symmetry on the level of the Hamiltonian is the most direct and transpar- ent way, as one can just read of this property immediately. Thereafter one can write down some new ��-symmetric Hamiltonians by means of simple deformations, i.e. replac- ing for instance the potential V(x) by V(x) f(ix), V(x) f(ixp), V(x) + f(ix) or V(x) + f(ixp), etc. with f being some arbitrary function. Clearly the Hamiltonians in (1) and (2) are of this type. Of course these new models are not guaranteed to have real spectra as the second property in (15) might be spoiled. Nonetheless, they have a high chance to describe non-dis- sipative physics and are potentially interesting. 3.1 ��-symmetric extensions for multi-particle systems Basu-Mallick and Kundu [39] were the first to write down some non-Hermitian extensions for some integrable many- -particle systems, i.e. the rational A�-Calogero models [40] �BK ii i k i k i k ii k p q g q q ig q q p � � � � � � � � � � 2 2 2 2 22 2 2 1 1 ( ) ~ ( )� (17) with g g,~ � �, q p, � ��� 1. There are some immediate ques- tions one may pose [41] with regard to the properties of �BK: i) How can one formulate �BK independently of the repre- sentation for the roots? ii) Can one generalize �BK to other potentials apart from the rational one? iii) Can one general- ize �BK to other algebras or more precisely Coxeter groups? iv) Is it possible to include more coupling constants and, in particular, v) Are the extensions still integrable? It turns out that the answers to all these questions become all quite simple when one realises that (17) corresponds in fact to the stan- dard Calogero model simply shifted in the momenta. This means the similarity transformation � is simply the transla- tion operator in p-space. In order to see this and to answer the above questions we ignore the confining term in (17) by taking � 0 and we re-write the Hamiltonian as � � � � � �� � � � � �� 1 2 1 2 2 2p g V q i p( ) � , (18) where � is now any root system invariant under Coxeter transformations, � � �� � � � ��1 2 ~ ( )g f q� , F x x( ) �1 and V x f x( ) ( )� 2 . We have also introduced coupling constants g g� �, ~ for each individual root. The Hamiltonians �� are meaningful for any representation of the roots and all Coxeter groups. For a specific choice of the representation for the roots, namely � � �i i i� � �1 for1 � �i � with � � �i j ij� � and the Coxeter group, i.e. A�, we recover the expression in (18). To establish the integrability of these models it is crucial to note the following not obvious property � � � � � � � 2 2 2 2 2� � � � � �� �s s l lg V q g V qs l ~ ( ) ~ ( ) � � (19) where �s, �l denotes the short and long roots, respectively. For the details of the proof of this identity we refer to [41]. As a consequence of (19), we may re-express �� in form of the usual Calogero Hamiltonian with shifted momenta together with some redefinitions of the coupling constants 46 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 � � � � � � � � � � � � � � � �� 1 2 1 2 2 2 2 2 2 2 ( ) � ( ), � ~ p i g V q g g gs s s � for � � � ! " � � s l l l lg g 2 2 2 � �~ . for (20) Therefore, upon redefining of the coupling constant, we may obtain �� by a similarity transformation as H h� � �� �1 Cal with � �� � �e x . The results of section 2.1 apply therefore and one may construct for instance the corresponding wavefunctions by �� � �� �1 Cal . Similarly one can establish integrability with the help of a Lax pair with a shifted momentum. One may verify that L p i H i g f q E M=m H i g f q E � � � � � � � � � ��( ) � ( ) � ( ) � � � � � � � � � � and �� � (21) fulfill the Lax equation � �� ,L L M� , upon the validity of the classical equation of motion resulting from (18), where the Lie algebraic commutation relations � � � � � � � � H H H E E E E H E E E i j i i, , , , , , , . � � � � �� � 0 � � � � � � ��� � � � � � are taken to be in the Cartan-Weyl basis, i.e. they are normal- ized as tr( )H Hi j ij� � , tr( )E E� �� �1 . The vector m can be expressed in terms of the structure constant ��� � and the po- tential in the usual fashion. We note that the Lax equation is ��-symmetric as �� : ,L L M M� � � . Naturally the con- served charges I Lk k� tr ( ) 2, notably the Hamiltonian I2, have the same property. Having established the integrability of the Calogero models one may address the question ii) and try to ex- tend these considerations to other potentials. Allowing now f x x( ) sinh�1 and f x x( ) sin�1 , we obtain the hyperbolic and elliptic case with V x f x( ) ( )� 2 . The integrability is guar- anteed by means of the same Lax pairs (21). However, when expanding the square in (20) the resulting Hamiltonian is not quite of the form (18) � � � � � � �� � � � � � �� 1 2 1 2 1 2 2 2 2p g V q i p� ( ) � , (22) because the identity (19) does not hold for the other poten- tials. This means that the Hamiltonians in (22) constitute non-Hermitian integrable extensions for Calogero-Moser- -Sutherland (CMS)-models for all crystallographic Coxeter groups, including, besides the rational, also trigonometric, hyperbolic and elliptic potentials. Dropping the last term would break the integrability for the non-rational potentials. 3.2 ��-symmetric deformations of the Korteweg deVries equation An even more popular integrable model than the CMS- -model is one having the Korteweg-de Vries (KdV) equation [42] as the equation of motion u uu ut x xxx� � � 0 . (23) This equation is known to remain invariant under x x� � , t t� � , u u� , i.e. it is ��-symmetric. By the same recipe out- lined above we may then carry out the following deformation u i iux x� � ( ) � with � � �, which was originally performed for the second term in [43] and for the third term in [44], leading to the equations u iu iu ut x xxx� � � �( ) � �0 � (24) and u uu i iu u iu ut x x xx x xxx� � � � � � � � � � � �( )( ) ( )1 02 2 1 , (25) respectively. For the model in (24) one can establish the following properties: the Galilean symmetry is broken, the model possesses two conserved quantities in terms of infinite sums and exhibits steady state solutions. However, it is un- clear how ��-symmetry can be utilized further. Instead (25), despite being more complicated, has some simpler proper- ties: it is Galilean invariant, possesses three simple conserved charges, exhibits steady state solutions, ��-symmetry can be utilized to explain the reality of the energy and it allows for a Hamiltonian formulation with non-Hermitian Hamiltonian density � � � � ��u iux 3 11 1 � � �( ) � . (26) Analogues of various different types of solutions of the KdV-equation have been studied in [43, 44]. No soliton solu- tions have been found and it seems unlikely that the models are integrable. 4 Conclusions We have demonstrated that ��-symmetry serves as a very useful guiding principle for constructing new interesting models, some of which even remain integrable. Being closely related to integrable models, these new models have ap- pealing features and deserve further investigation. 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Andreas Fring phone: +44 (0)20 7040 4123 e-mail: a.fring@city.ac.uk Centre for Mathematical Science City University Northampton Square London EC1V 0HB, United Kingdom (and Department of Physics University of Stellenbosch 7602 Matieland, South Africa) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 49 Acta Polytechnica Vol. 47 No. 2–3/2007