Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
aa r X i v : . [ h e p - t h ] S e p Interface between Hermitian andnon-Hermitian Hamiltonians in a modelcalculation
H. F. Jones † Physics Department, Imperial College, London SW7 2AZ, UKNovember 6, 2018
Abstract
We consider the interaction between the Hermitian world, repre-sented by a real delta-function potential − αδ ( x ), and the non-Hermitianworld, represented by a PT-symmetric pair of delta functions withimaginary coefficients iβ ( δ ( x − L ) − δ ( x + L )). In the context of stan-dard quantum mechanics, the effect of the introduction of the imagi-nary delta functions on the bound-state energy of the real delta func-tion and its associated wave-function is small for L large. However,scattering from the combined potentials does not conserve probabilityas conventionally defined. Both these problems can be studied insteadin the context of quasi-Hermiticity, whereby quantum mechanics isendowed with a new metric η , and consequently a new wave-functionΨ( x ), defined in terms of the original wave-function ψ ( x ) by means of η . In this picture, working perturbatively in β , the bound-state wave-function is actually unchanged from its unperturbed form for | x | ≪ L .However, the scattering wave-function, for | x | ≫ L , is changed in a sig-nificant manner. In particular there are incoming and outgoing waveson both sides of the potential. One can then no longer talk in termsof reflection and transmission coefficients, but the total right-movingflux is now conserved. † e-mail: [email protected] Introduction
Since the resurgence of interest in non-Hermitian Hamiltonians initiated bythe paper of Bender and Boettcher[1] the subject has gone through severalstages. First there was the search for soluble non-Hermitian Hamiltonianswith real spectra (see, e.g. [2]). The next stage concerned the introduc-tion of a new Hilbert-space metric in order to obtain positive probabili-ties and so regain a physical interpretation of the theory. This was firstdone in
P T -symmetric theories[3] by introducing the so-called C operatorto form the metric CP T . A more general framework was formulated byMostafazadeh[4], which among other things established the connection toearlier work by Scholtz et al.[5] and showed[6] that such a Hamiltonian H was related by a similarity transformation to an equivalent Hermitian Hamil-tonian h . Subsequent work showed how the metric η could be constructed,sometimes exactly, but more typically in perturbation theory for a variety ofmodels[7, 8, 9].More recently some attention has been given to situations where a non-Hermitian system interacts with the world of Hermitian quantum mechanics.For example, Ref. [10] examined a non-Hermitian analogue of the Stern-Gerlach experiment in which the role of the intermediate inhomogeneousmagnetic field flipping the spin is taken over by an apparatus described bya non-Hermitian Hamiltonian. This type of set-up has been further elab-orated by Assis and Fring[11] and Guenther al.[12], and Mostafazadeh[13]has emphasized that the effect relies on non-unitarity in some guise or an-other (see also [14]). The subject has also taken an experimental turn in theform of optical lattices whose refractive index can be tailored to make them P T -symmetric (see, for example, [15]).An earlier attempt at understanding the conceptual issues involved inscattering from a non-Hermitian potential was given in Ref. [16]. There weprimarily considered a single delta-function at the origin with a complexcoefficient. Here unitarity is not conserved within the framework of conven-tional quantum mechanics, but if one instead constructs the metric η and thecorresponding transformed wave-function Ψ (see Eq. (20) below) unitarity isrestored but at the price of a drastic change in the physical picture, wherebythere are now incoming and outgoing waves on both sides of the potential.A somewhat less drastic, but still significant, change is found by Znojil[17]in a discretized model of scattering, where the metric does not mix up in-coming and outgoing waves, but instead changes the normalization of the2ux on either side of the scattering centre. In Ref. [16] we touched on thepotential model that we address in the present paper, but only in the contextof conventional quantum mechanics. On the basis of those calculations wespeculated that the treatment of bound-state problems should be essentiallyunaffected by the introduction of distant non-Hermitian scattering poten-tials, and that it is only when the system physically interacts with thosepotentials that a drastic change in the formalism is required, if indeed thenon-Hermitian potentials are regarded as fundamental rather than effective.In the present paper we return to that model, which we are now able totreat in the quasi-Hermitian picture as well, thanks to a general prescrip-tion due to Mostafazadeh[18] for calculating η perturbatively in the case ofa superposition of delta functions. The model is introduced in Section 2,where it is treated in the framework of conventional quantum mechanics. Aswas already seen in Ref. [16], the bound-state energy and wave-function areaffected only by exponentially small terms when L → ∞ , but the scatter-ing does not preserve unitarity, with R + T = 1. In Section 3 we insteadtreat the problem in the quasi-Hermitian framework, calculating the metricto first order in β but to all orders in α . For the bound state we find thatthe new wave-function Ψ is actually equal to the old wave-function ψ in theabsence of the non-Hermitian part of the Hamiltonian. For the scatteringproblem we find the same type of wave-function that we previously found forthe single complex delta function, whereby Ψ contains an incoming wave onthe right-hand side, in contrast to ψ . Finally, in Section 4, we discuss thesignificance of these results. The Hamiltonian we analyze in this paper has a potential consisting of threedelta functions: H = p − αδ ( x ) + iβ ( δ ( x − L ) − δ ( x + L )) . (1)The first component of the potential, − αδ ( x ), is a delta function based onthe origin with a real, negative coefficient. When β = 0 this Hermitian partof the potential supports a single bound state with energy E = − κ , where κ = α . The second component proportional to β consists of two deltafunctions based at x = ± L with imaginary coefficients ± iβ , designed to be P T -symmetric and have real energy eigenvalues. What is at issue is how the3ntroduction of this non-Hermitian piece of the Hamiltonian affects both thescattering wave-function and the bound-state energy and wave-function. Ofparticular interest is the case when L is large, i.e. when the non-Hermitianpieces are distant from the Hermitian potential based at the origin. Let us start by setting β = 0. As already stated, there is then a single boundstate, with (unnormalized) wave-function ψ = e − κ | x | , (2)where κ = α in order to satisfy the continuity condition ψ ′ (0 + ) − ψ ′ (0 − ) = αψ (0) . (3)Let us now repeat this calculation for β = 0. The bound-state wave-function then has the ( P T -symmetric) form ψ ( x ) = W e κx x < − LU e κx + V e − κx − L < x < U ∗ e − κx + V ∗ e κx < x < LW ∗ e − κx L < x (4)with ψ ( − x ) = ψ ∗ ( x ). Applying the continuity conditions ψ ( − L + ) = ψ ( − L − ), ψ ′ ( − L + ) − ψ ′ ( − L − ) = − iβψ ( − L ) at x = − L we find U = (1 − i ˜ β ) WV = i ˜ βe − κL W, (5)where ˜ β = β/ (2 κ ). The continuity conditions at x = 0 are U + V = U ∗ + V ∗ κ [( U + U ∗ ) − ( V + V ∗ )] = α ( U + V ) . (6)The first of these gives h (1 − i ˜ β ) + i ˜ βe − κL i W = h (1 + i ˜ β ) − i ˜ βe − κL i W ∗ , (7)which can be satisfied by taking W = (1 + i ˜ β ) − i ˜ βe − κL , (8)4p to an overall real normalization constant. On substitution into the secondcontinuity condition we find, after some algebra, the eigenvalue equation2 κ h β (1 − e − κL ) i = α h β (1 − e − κL ) i , (9)which, after some additional manipulation, can be written as˜ α = 1 + (1 + ˜ α ) ˜ β e − κL (1 − e − κL )1 + ˜ β (1 − e − κL ) , (10)where ˜ α = α/ (2 κ ). In this form it is clear that (i) κ → α as L → ∞ , and (ii)the first correction to κ is of order β . It is interesting that the wave-functiondoes not depend explicitly on α , only through the relation between κ and α . Again let us start with β = 0. The scattering wave-function, in the situ-ation where a plane wave comes in from the left and is either reflected ortransmitted at the delta-function potential, is of the form ψ ( x ) = (cid:26) Ae ikx + Be − ikx x < e ikx x > A and B are determined by the continuity condition (3) as A = 1 − i ˆ αB = i ˆ α, (12)where ˆ α = α/ (2 k ), giving reflection ( R ) and transmission ( T ) coefficients T = 1 / | A | = 1 / (1 + ˆ α ) R = | B | / | A | = ˆ α / (1 + ˆ α ) . (13)Since the potential for β = 0 is real, the scattering is unitary, with R + T = 1.For β = 0 the wave-function has the general form ψ ( x ) = Ae ikx + Be − ikx x < − LCe ikx + De − ikx − L < x < Ee ikx + F e − ikx < x < Le ikx L < x (14)5orking from the right, we first apply the continuity conditions at x = L ,to obtain E = 1 − ˆ βF = ˆ βe ikL , (15)where ˆ β = β/ (2 k ). Then, applying the continuity conditions at x = 0, weobtain C = (1 − i ˆ α )(1 − ˆ β ) + i ˆ α ˆ βe ikL D = (1 + i ˆ α ) ˆ βe ikL + i ˆ α (1 − ˆ β ) . (16)Finally, applying the continuity conditions at x = − L we obtain, after somealgebra, A = (1 − ˆ β )(1 − i ˆ α ) − i ˆ α ˆ β e ikL + ˆ β (1 + i ˆ α ) e ikL (17) B = ˆ β (1 − ˆ β )[(1 + i ˆ α ) e ikL − (1 − i ˆ α ) e − ikL ] + i ˆ α [ ˆ β + (1 − ˆ β ) ] . Note that A only involves ˆ β , whereas B involves ˆ β linearly. The coefficientsare at most linear in α . As expansions in β we have A = 1 − i ˆ α + O ( β ) (18) B = i ˆ α + 2 i ˆ β [sin 2 kL − α sin kL ] + O ( β ) . Thus the transmission and reflection coefficients are T = 1 / (1 + ˆ α ) R = ( ˆ α + 4 ˆ α ˆ β sin 2 kL ) / (1 + ˆ α ) (cid:27) + O ( β , α β ) . (19)Clearly unitarity, as conventionally defined, is violated in this process. The Hamiltonian has been specifically constructed to be
P T symmetric. Insuch cases, as discussed in the introduction, we can in principle introducea positive-definite metric operator[4, 19] η = e − Q with respect to which H is quasi-Hermitian: H † = ηHη − . Observables are those represented byquasi-Hermitian operators A such that A † = ηAη − . The original positionoperator x does not fall into this category: instead the observable of position6s X ≡ ρxρ − , where ρ = η = e − Q . Consequently[20, 21], in this picturethe relevant wave-function is not ψ ( x ) ≡ h x | ψ i , butΨ( x ) ≡ h x | Ψ i = h x | ρ | ψ i = Z ρ ( x, y ) ψ ( y ) dy. (20)Let us therefore attempt to construct Ψ for the present Hamiltonian. Un-fortunately this cannot be done exactly, but Mostafazadeh[18] has deviseda general method for constructing a series expansion for η in the couplingconstants of a series of delta functions. This method is based on expressingthe condition of quasi-Hermiticity of the Hamiltonian as a partial differentialequation for η ( x, y ), and converting it into an integral equation that can besolved iteratively.In some more detail, for the Hamiltonian H = p + V ( x ), the integralequation for η ( x, y ) takes the form η ( x, y ) = u ( x, y ) + ( Kη )( x, y ) , (21)where u ( x, y ) ≡ u + ( x − y )+ u − ( x + y ) is the general solution of the differentialequation ( − ∂ x + ∂ y ) u ( x, y ) = 0, and K is defined by( Kη )( x, y ) = (cid:18)Z y dr V ( r ) Z x + y − rx − y + r ds + Z x ds V ∗ ( s ) Z x + y − sy − x + s dr (cid:19) η ( s, r )(22)In Ref. [18] the equation (21) was written in the form η = (1 − K ) − u, (23)which in principle can be expanded in K , i.e. as a simultaneous expansion inthe coefficients of the delta functions. However, in the present case we need todo something slightly different, because we are thinking of β as a perturbativeparameter, but not α . Thus we need to split K up into K = K α + K β . Thenwe write Eq. (21) as η = u + ( K α + K β ) η. (24)Now for β = 0 the Hamiltonian is Hermitian, so we want η ( x, y ) = δ ( x − y ).This means that to order β we should take u = (1 − K α ) δ . In principle wecould also add to u a term βw , where w is a solution of the homogenousequation ( ∂ x − ∂ y ) w ( x, y ) = 0. However, it turns out that such a term is notrequired. So η = (1 − K α ) δ + ( K α + K β ) η, (25)7o that (1 − K α ) η = (1 − K α ) δ + K β η, (26)with solution η = δ + 11 − K α K β η . (27)To O ( β ), which is as far as we will take the calculation, this reduces to η = δ + 11 − K α K β δ = δ + (1 + K α + K α + . . . ) K β δ . (28)Since we would like to treat β perturbatively, but not α , it is extremely fortu-nate that the higher powers of K α in this equation do not in fact contribute,as is shown in Section 3.2. K β As Mostafazadeh has shown[18] (with m = , ~ = 1), the action of K β onthe delta function δ ( x − y ) is( K β δ )( x, y ) = 12 iβ [ θ ( x + y − L ) − θ ( x + y + 2 L )] ε ( y − x ) , (29)where ε ( z ) is the sign function ε ( z ) ≡ sgn( z ). Thus, to order β we have η ( x, y ) = δ ( x − y ) + x + y < − L iβε ( x − y ) − L < x + y < L L < x + y ≡ δ ( x − y ) − βQ ( x, y ) (30) Let us now calculate the effect of ρ ( x, y ) = δ ( x − y ) − βQ ( x, y ) on thebound-state wave-function Ψ( x ). From Eq. (20) it is given, to this order, byΨ( x ) = ψ ( x ) + 14 iβ Z L − x − L − x ε ( x − y ) ψ ( y ) dy = ψ ( x ) − iβ (cid:20)Z L − xx ψ ( y ) dy − Z x − L − x ψ ( y ) dy (cid:21) , (31)8or 0 < x < L . Here, since we are working to O ( β ), we can take ψ ( y ) = e − κ | y | in the integrands. ThenΨ( x ) = (1 + i ˜ βe − κL ) e − κx − i ˜ βe − κL e κx − iβ (cid:20)Z L − x e − κy dy − Z x e − κy dy − Z − L − x e κy dy (cid:21) (32)= (cid:18) i ˜ βe − κL (cid:19) e − κx − i ˜ βe − κL e κx + i ˜ β (1 − e − κx ) + O ( β ) . The last term is not something we expect at all, but before jumping toconclusions we should await the calculation of the K α K β contribution, which,because it involves α in the combination ˜ α ≡ α/ (2 κ ) = 1+ O ( β ), is of exactlythe same order. We are now concerned with the scattering wave-function for large | x | . For x > x > L (so that 2 L − x < − L ), at whichpoint the wave-function, Ψ > settles down to its asymptotic form, namelyΨ > ( x ) = e ikx + 14 iβ Z L − x − L − x ε ( x − y )[(1 − i ˆ α ) e iky + i ˆ αe − iky ] dy = e ikx + 14 iβ Z L − x − L − x [(1 − i ˆ α ) e iky + i ˆ αe − iky ] dy (33)= (1 − ˆ α ˆ β sin 2 kL ) e ikx + i ˆ β (1 − i ˆ α ) sin 2 kL e − ikx + O ( β ) . For x < − L (so that − L − x > L ) we instead getΨ < ( x ) = Ae ikx + Be − ikx + 14 iβ Z L − x − L − x ε ( x − y ) e iky dy = Ae ikx + Be − ikx − iβ Z L − x − L − x e iky dy (34)= (1 − i ˆ α ) e ikx + i [ ˆ α + ˆ β sin 2 kL − α ˆ β sin kL )] e − ikx + O ( β ) . Note that the physical picture has completely changed, because we haveincoming and outgoing waves on both sides. This is the rather drastic mod-ification noted in Ref. [16]. Some such modification at infinity is clearly9ecessary if we are to restore unitarity in this picture. In a recent discretizedmodel of scattering studied by Znojil[17] the modification is instead a changein the normalization of the fluxes on either side of the scattering centre. Inthe present model we can check unitarity by comparing the net right-movingfluxes Φ on each side. Unitarity is indeed restored to this order, becauseΦ > = Φ < = 1 − α ˆ β sin 2 kL + O ( α β, β ) . (35)Note that we are only allowed to calculate the fluxes using the standardformula in regions where the equivalent Hermitian Hamiltonian h is simply p . Otherwise[16] the conservation of probability takes a non-local forminvolving an integral over h ( x, y ). K α K β According to Eq. (28) we need to calculate K α Q . In general[18], the effectof K α on u ( x, y ) is( K α u )( x, y ) = − α (cid:20) θ ( y ) Z x + yx − y dr u ( r,
0) + θ ( x ) Z y + xy − x ds u (0 , s ) (cid:21) . (36)Recall, cf. Eq. (30), that Q ( x, y ) = iε ( y − x )[ θ ( x + y + 2 L ) − θ ( x + y − L )].Thus ( K α Q )( x, y ) = 14 iαθ ( y ) Z x + yx − y dr ε ( r )[ θ ( r + 2 L ) − θ ( r − L )] − iαθ ( x ) Z y + xy − x ds ε ( s )[ θ ( s + 2 L ) − θ ( s − L )] . (37)In principle we need to calculate the effects of K nα arising from the expansionof Eq. (28). However, a surprising and welcome result is that these vanishfor n >
1. Thus, in the calculation of K α Q according to Eq. (36), we need( K α Q )( r,
0) = − iαθ ( r ) Z r − r du ε ( u )[ θ ( u + 2 L ) − θ ( u − L )]= 0 by symmetry . (38)Similarly ( K α ¯ Q )(0 , s ) = 0. So in fact all higher order terms in Eq. (28) areabsent. 10 .2.1 Bound-state wave-function Here, since we are concerned with the limit as L → ∞ , we need to evaluateEq. (37) for 0 < x ≪ L . A similar analysis will apply to x <
0, but thisis easily obtained by
P T symmetry. First it is easy to see that for a non-zero result x and y must have opposite signs. So in the present case y < y + x and y − x . The net result is( K α Q )( x, y ) = − iα y > y > − x − x − x > y > − L + x − (2 L + y + x ) − L + x > y > − L − x , (39)giving a correction to the bound-state wave function∆Ψ = 18 iαβ (cid:26) Z − x y − x Z − x − L + x − Z − L + x − L − x (2 L + y − x ) (cid:27) e κy dy = 12 i ˜ α ˜ β (cid:2) e − κL ( e κx − e − κx ) − − e − κx ) (cid:3) . (40)When added to Ψ of Eq. (32), and remembering that ˜ α = 1 + O ( β ), weobtain the remarkably simple resultΨ( x ) = e − κx + O ( β ) , (41)which to this order is equal to the original undisturbed wave function ψ ( x )of Eq. (2) for x > For the corrections to the scattering wave-function we need to evaluateEq. (37) for | x | ≫ L . Let us first consider x ≫ L , in order to obtainthe correction ∆Ψ > to Ψ > . Again, for a non-zero result y must be negative,so that( K α Q )( x, y ) = − iα Z y + xy − x ds ε ( s )[ θ ( s + 2 L ) − θ ( s − L )] . (42)Here the lower limit, y − x , is less than − L , and there are again two possi-bilities depending on the position of the upper limit. The net result is( K α Q )( x, y ) = 14 iα x + y − L < x + y < L − ( x + y + 2 L ) − L < x + y <
00 otherwise , (43)11iving ∆Ψ > = − αβ (cid:20)Z − x − L − x dy ( x + y − L ) ψ < ( y ) − Z L − x − x dy ( x + y + 2 L ) ψ < ( y ) (cid:21) (44)where ψ < ( y ) = (1 − i ˆ α ) e iky + i ˆ αe − iky . The result of this calculation is∆Ψ > = 2 ˆ α ˆ β sin kL (cid:2) (1 − i ˆ α ) e − ikx + i ˆ αe ikx (cid:3) , (45)giving the corrected value of Ψ > asΨ > ( x ) = (1 − ˆ α ˆ β sin 2 kL + 2 ˆ α ˆ β sin kL ) e ikx + i ˆ β (1 − i ˆ α )(sin 2 kL + 2 ˆ α sin kL ) e − ikx . (46)Now let us take x ≪ − L , so that y must be positive. The expression for K α Q turns out to be the same as that given in Eq. (39). Thus∆Ψ < = − αβ (cid:20)Z − x − L − x dy ( x + y − L ) ψ > ( y ) − Z L − x − x dy ( x + y + 2 L ) ψ > ( y ) (cid:21) (47)where ψ > ( y ) = e iky . Hence∆Ψ < = 2 ˆ α ˆ β sin kLe − ikx , (48)giving the corrected value of Ψ < asΨ < ( x ) = (1 − i ˆ α ) e ikx + i [ ˆ α + ˆ β sin 2 kL − α ˆ β sin kL ) e − ikx . (49)From Eqs. (46) and (49) we obtain the fluxesΦ > = Φ < = 1 − α ˆ β (sin 2 kL − α sin kL ) + O ( β ) . (50)Compared with Eq. (35), we now have the explicit O ( α β ) terms, and theresult is correct to the order shown. 12 Discussion
Let us now consider the conceptual issues raised by these calculations. Firstit should be emphasized that we are concerned here with quasi-HermitianHamiltonians, that is, Hamiltonians that can be related by a similarity trans-formation to a Hermitian Hamiltonian. It is only for such Hamiltonians thatwe can attempt to construct the metric η and to look at the situation inthe quasi-Hermitian framework. For generic non-Hermitian Hamiltoniansno such framework is available, and one is bound to treat them as effectiveHamiltonians within the standard framework of quantum mechanics. In thatcase one would simply perform an analysis similar to that of Section 2 andaccept that unitarity is not conserved, essentially because we are dealing witha subsystem of a larger system whose physics has not been taken fully intoaccount.For the potential we have chosen we are able to consider both bound andscattering states. The bound state is the simpler to consider. The standardquantum mechanical analysis shows that the introduction of the perturbing P T -symmetric delta-function potentials does not significantly modify eitherthe bound-state energy or the wave-function if these potentials are sufficientlydistant. The analysis of Section 3, in particular the final result of Eq. (41),shows that this remains the case within the quasi-Hermitian framework, withthe new bound-state wave-function Ψ( x ) being identical to the original wave-function ψ ( x ) for large L . This is a new and reassuring result, which we wereunable to address in Ref. [16], where the potential did not support a boundstate. The general message we would like to draw from this is that a localizedphysical Hermitian system is not significantly affected by the introductionof distant non-Hermitian potentials, and may be treated in the frameworkof standard quantum mechanics without the necessity of introducing a newmetric.The real conceptual problems arise for the scattering states. The stan-dard quantum mechanical analysis shows that unitarity, as conventionallydefined, is not conserved. In the quasi-Hermitian framework, however, onecan hope that a modified form of unitarity is in fact conserved, as is indeedborne out by the calculations of Section 3. However, this involves a fairlydrastic redefinition of asymptotic states, a non-local effect, given the finitesupport of the perturbing P T -symmetric potentials. The crux is the differ-ence between x , the coordinate parameter in terms of which the Hamiltonianis originally defined, and X , its quasi-Hermitian counterpart, defined by the13on-local relation X ≡ ρxρ − . The former, x , is Hermitian and thereforean observable, in the standard framework of quantum mechanics, while X isnot. Conversely, in the quasi-Hermitian picture, X is quasi-Hermitian andan observable, while x is not. The argument of the first wave-function ψ ( x )is the eigenvalue of x , while that of Ψ( x ) is the eigenvalue of X .What we have done in this paper is transform an initial scattering set-up defined in terms of x , and then consider the corresponding picture inthe quasi-Hermitian framework in terms of X . The initial scattering set-uphad plane waves entering from the left and then being either reflected ortransmitted, with probability not being conserved. As we have seen, thecorresponding quasi-Hermitian picture is that the newly-defined probabilityis indeed conserved, but that waves now enter from both left and right. Analternative mathematical possibility would be to set up a scattering situationin which Ψ > has only outgoing waves and then work backwards to construct ψ , which would undoubtedly have waves entering from both left and right.In either case, the physical picture changes drastically when going from onepicture to the other † .In the author’s opinion, the only satisfactory resolution of this dilemmais to treat the non-Hermitian scattering potential as an effective one, andwork in the standard framework of quantum mechanics, accepting that thiseffective potential may well involve the loss of unitarity when attention isrestricted to the quantum mechanical system itself and not its environment.This is indeed the attitude taken in a recent paper by Berry [23], where var-ious intensity sum rules are derived for diffraction off P T -symmetric opticallattices. There it is taken for granted that the intensities are given by | ψ | . Itis true that these are classical calculations, but because of the correspondenceprinciple the same thing would apply in quantum mechanics. † In a recent paper, ref. [22], Znojil has constructed certain discrete matrix models ofscattering in which in and out states are not mixed in the quasi-Hermitian picture, butinstead the definition of the flux differs on the left and right of the scattering centre. Thisis a somewhat less drastic change, but still involves a departure from standard quantummechanics at large distances. eferences [1] C. M. Bender and S. Boettcher, Phys. Rev. Lett. (1998) 5243.[2] G. Levai and M. Znojil, J. Phys. A: Math. Gen. (2000) 7165.[3] C. M. Bender, D. C. Brody and H. F. Jones, Phys. Rev. Lett. (2002)270401; ibid. (2004) 119902 (E).[4] A. Mostafazadeh, J. Math. Phys. (2002) 205.[5] F. Scholtz, H. Geyer and F. Hahne, Ann. Phys. (1992) 74.[6] A. Mostafazadeh, J. Phys. A: Math. Gen. (2003) 7081.[7] H. F. Jones, J. Phys. A: Math. Gen. (2005) 1741.[8] A. 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