A universal form of complex potentials with spectral singularities
aa r X i v : . [ n li n . PS ] O c t A universal form of complex potentials withspectral singularities
Dmitry A. Zezyulin & Vladimir V. Konotop ITMO University, St. Petersburg 197101, Russia Departamento de F´ısica and Centro de F´ısica Te´orica e Computacional, Faculdadede Ciˆencias, Universidade de Lisboa, Campo Grande 2, Edif´ıcio C8, Lisboa 1749-016,PortugalE-mail: [email protected]
Abstract.
We establish necessary and sufficient conditions for complex potentialsin the Schr¨odinger equation to enable spectral singularities (SSs) and show that suchpotentials have the universal form U ( x ) = − w ( x ) − iw x ( x ) + k , where w ( x ) is adifferentiable function, such that lim x →±∞ w ( x ) = ∓ k , and k is a nonzero real. Wealso find that when k is a complex number, then the eigenvalue of the correspondingShr¨odinger operator has an exact solution which, depending on k , represents acoherent perfect absorber (CPA), laser, a localized bound state, a quasi bound state inthe continuum (a quasi-BIC), or an exceptional point (the latter requiring additionalconditions). Thus, k is a bifurcation parameter that describes transformations amongall those solutions. Additionally, in a more specific case of a real-valued function w ( x )the resulting potential, although not being PT symmetric, can feature a self-dualspectral singularity associated with the CPA-laser operation. In the space of the systemparameters, the transition through each self-dual spectral singularity corresponds to abifurcation of a pair of complex-conjugate propagation constants from the continuum.The bifurcation of a first complex-conjugate pair corresponds to the phase transitionfrom purely real to complex spectrum. Keywords : non-Hermitian potentials, spectral singularities, lasing, coherent perfectabsorption
1. Introduction
Singularities of the spectral characteristics of non-Hermitian operators, alias spectralsingularities (SSs), were introduced in mathematical literature more than six decadesago [1] and were well studied since then [2, 3, 4]. Independently on these studies, therewere appearing physical examples of absorbers [5, 6, 7] (see also [8]) and lasers [7]of systems possessing SSs (although without direct reference on the notion of SS).The close relation between SSs, and the physical concepts of coherent perfect absorber(CPA), laser, and zero width resonances were established more recently in a series of universal form of complex potentials with spectral singularities M ( k ) depending on the wavenumber k , when real zeros of the matrixelement M ( k ) determine SSs [9, 10]. This approach indicates that for a given localizedcomplex potential the existence of a SS is a delicate property, requiring precise matchingof the physical parameters ensuring a real zero of a complex function M ( k ). Hence anumber of free parameters of the potential must be big enough in order to ensure theexistence of a SS. In the meantime, it turns out that in spite of these, sometimes severe,constraints, a number of potentials that support SSs with the prescribed properties(these including the wavelength, the order of a SS, the number of SSs, etc.) is infinitelylarge and can be constructed in an algorithmic way [16, 17, 15]. The remaining questions,however, are whether the structure of all such complex potentials, supporting SSs, andwhether the field structure corresponding to those potentials have something in commonor not.In this paper we give positive answer to both above questions. More specifically,subject to some quite weak (physically) constraints, we establish necessary (section 2)and sufficient (section 3) conditions on the form of a complex potential that featuresSSs. This universal form reads U ( x ) = − w ( x ) − iw x ( x ) + k , where U ( x ) is thecomplex potential, k is a nonzero real, and w ( x ) is a differentiable function whichfeatures asymptotic behaviour lim x →±∞ w ( x ) = ∓ k and satisfies certain additional(not very restrictive) requirements. Our proofs are based on a specific representationof the SS solutions which allows for describing the parametric transformation of thecomplex potential making it supporting other types of solutions including bound states,quasi-bound states in continuum, and exceptional point solutions (4). Furthermore, ina more specific situation of real-valued function w ( x ) we demonstrate that the foundSS solution can coexist with another, self-dual spectral singularity which correspondsto the combined CPA-laser operation (section 5). In concluding section 6 we provide anoutlook towards possible practical implementation of our findings and some promisinggeneralizations of the presented theory.
2. Universal form of a complex potential resulting in a spectral singularity
Our main goal is to study some general properties of special solutions of a one-dimensional Schr¨odinger equation − ψ ′′ + U ( x ) ψ = k ψ, (1)where U ( x ) is a spatially localized complex-valued potential, i.e.lim x →±∞ U ( x ) = 0 , (2) universal form of complex potentials with spectral singularities k is a spectral parameter (hereafter a prime stands for a derivative with respectto x ). More specifically, we are interested in solutions ψ ( x ) of (1) which satisfy theconditions as follows:(i) the function ψ ( x ) is a continuously differentiable, i.e., ψ ∈ C ( R );(ii) for a given value of the spectral parameter k = k ∈ R , the function ψ ( x ) ischaracterized by the asymptotic behaviourlim x →±∞ [ ψ ( n )0 ( x ) − ( ± ik ) n e ± ik x ρ ± ] = 0 , ρ ± = 0 , n = 0 , , , (3)where the superscript “( n )” denotes the n th derivative in x : ψ ( n )0 ( x ) ≡ d n ψ /dx n , and ρ ± ∈ C are nonzero complex constants. A solution satisfying the formulated constraintswill be referred to as a SS-solution. Requirements (3) imply that a SS-solution ψ ( x )describes a laser for k > k <
0. Taking into that the 2 × M ( k ) links the Jost solutions of any localized potential (see e.g. [18]),the definition of SS-solution given above is consistent with the standard definition of aSS. Namely, k is a SS if M ( k ) = 0 (hereafter M ij ( k ) with i, j = 1 , M ( k )).Consider some function ψ ( x ) that has the above properties (i) and (ii) and solvesequation (1). Then one can show that subject to some additional conditions such ψ ( x )requires the potential U ( x ) to have the asymptotic behavior (2), as well as to admita special representation, given by equation (12) below. Starting with these additionalconditions, we assume that ψ ( x ) has only a finite number N of simple roots (i.e. rootsof multiplicity one) denoted by x j ( j = 1 , , .., N ) and ordered as −∞ < x < x < . . . < x N < ∞ . (4)Then it follows from Taylor’s theorem that at x → x j ψ ( x ) = ψ ′ ( x j )( x − x j ) + o ( x − x j ) , ψ ′ ( x ) = ψ ′ ( x j ) + o (1) ,ψ ′ ( x j ) = 0 . (5)For the next consideration, it is convenient to introduce intervals on the real axiswhere ψ ( x ) = 0 (see figure 1) I = ( −∞ , x ) , I = ( x , x ) , . . . I N − = ( x N − , x N ) , I N = ( x N , ∞ ) , (6)as well as the set I of all point of the real axis where ψ ( x ) is nonzero: I = I ∪ I ∪ · · · ∪ I N . (7)For x ∈ I one can define the function w ( x ) = i ψ ′ ( x ) ψ ( x ) (8)which is continuous in I and has the following additional properties (see figure 1)lim x →±∞ w ( x ) = ∓ k , (9) w ( x ) = ix − x j + o (cid:18) x − x j (cid:19) as x → x j . (10) universal form of complex potentials with spectral singularities Figure 1.
Illustration to theorems 2.1 and 3.1. Points x < x < . . . < x N divide thereal x -axis into intervals I , I , . . . , I N . Each interval contains an arbitrarily chosenpoint X , X , . . . , X N . Additionally, a schematics of singularities and asymptoticbehaviour of | w ( x ) | are indicated. Additionally, let us also assume that in each interval I j there exists the second derivative ψ ′′ ( x ) which is bounded in each closed subset of I . Then w ( x ) is differentiable in I , andits derivative w ′ ( x ) is also bounded in each closed subset of I . Asymptotic behavior ofthe SS-solution ψ required in (3) implies thatlim x →±∞ w ′ ( x ) = 0 . (11)Then, for ψ ( x ) to be a solution of Schr¨odinger equation (1), the complex potential inthis equation must have the form U ( x ) = − w ( x ) − iw ′ ( x ) + k . (12)For k = 0 and real-valued function w ( x ), the potentials of this type have been proposedin earlier works [19, 20] for obtaining complex potentials with real spectra. It followsfrom (3) that the potential introduced through (12) has zero asymptotic behaviour, asprescribed by (2), and is a bounded function in any closed interval belonging to I .Since the starting point of the above analysis was a SS-solution, now we canformulate a necessary condition for a complex potential U ( x ) to result in a SS of therespective Schr¨odinger operator. Theorem 2.1 (Necessary condition) . Let a function ψ ∈ C ( R ) have at most a finitenumber of simple roots x j ordered as in (4), have asymptotic behaviour (3), and in theset I defined by (7) have the second derivative ψ ′′ ( x ) which is bounded in each closedsubset of I . If ψ ( x ) solves Schr¨odinger equation (1) with a complex potential U ( x ) ,then (i) U ( x ) is bounded in any closed sub-interval of I and vanishes at | x | → ∞ , (ii)for x ∈ I this potential allows for representation (12) with the function w ( x ) havingasymptotics (9), and (iii) the function w ( x ) is defined by (8). We notice that in the isolated points x j / ∈ I the potential U ( x ) is allowed to havesingularities. If however, a function ψ is different from zero on the whole real axis, i.e. universal form of complex potentials with spectral singularities ψ ( x ) = 0 for x ∈ R , then neither w ( x ) nor U ( x ) have singularities. In this last case,solving (8) with respect to ψ , we find that the solution corresponding to the spectralsingularity of potential (12) can be expressed directly through the function w ( x ), as ψ ( x ) = ρ exp (cid:20) − i Z xx w ( ξ ) dξ (cid:21) , (13)where x and ρ are arbitrary constants ( ρ = 0). Recently, some SS-solutions having thefrom (13) and related to the potential (12) were discussed in [21].The potential U ( x ) determined in Theorem 2.1 may have discontinuities. For thesake of illustration, let us show that the known example of a complex rectangularpotential [9, 15]: U ( x ) = ( k − κ , at | x | ≤
10 at | x | > , (14)where κ is a complex number, for the parameters enabling a SS-solution can berepresented in the from (12). Indeed, let k be a SS. This means that either κ issuch that for a real k we have k = iκ tan κ , and then we compute the an SS-solution ψ ( x ) = ( κe ik ( | x |− | x | ≥ κ cos( κx ) / cos κ | x | ≤ w ( x ) = ( ∓ k ± x > − iκ tan( κx ) | x | ≤ , (16)or k = − iκ cot κ and we obtain an odd SS-solution ψ ( x ) = ( ± κe ik ( | x |− ± x > κ sin( κx ) / sin κ | x | ≤ w ( x ) function w ( x ) = ( ∓ k ± x > iκ cot( κx ) | x | ≤ . (18)Since the odd solution has zero at x = 0, the function w ( x ) has a singularity at thispoint, which however correspond to a bounded value U (0) of the potential (as this isschematically illustrated in Fig. 1).
3. Construction of solutions corresponding to spectral singularities.Sufficient condition.
In the previous section, we have demonstrated that if Schr¨odinger equation (1) has aSS solution ψ ( x ) obeying certain additional properties, then the potential U ( x ) in thisequation admits a representation in the form (12), where function w ( x ) can be foundfrom the solution ψ ( x ). In this section we address a converse situation. Suppose thatfunction w ( x ) is known, and the potential U ( x ) has the form (12). Our goal now is universal form of complex potentials with spectral singularities w ( x ). In fact, this solutioncan be generalized on the case when w ( x ) has discontinuities of a certain type. As weshall demonstrate below, in this situation solution ψ ( x ) should be defined piecewise oneach continuity interval of w ( x ), and the amplitude of ψ ( x ) vanishes at the points ofdiscontinuity.Turning to rigorous formulation, we consider a function w ( x ) which is definedpiecewise in intervals I j ( j = 0 , , . . . N ) introduced in (6) (we notice that N = 0corresponds to a continuous function w ( x )). More specifically, we assume that (i)for x ∈ I j , w ( x ) ≡ w j ( x ), where each function w j ( x ) is continuous and piecewisedifferentiable in I j ; (ii) in the points x j the function w ( x ) has discontinuities defined bythe representations w ( x ) = ix − x + w − ( x ) ,w ( x ) = ix − x + w +1 ( x ) = ix − x + w − ( x ) , ... w N − ( x ) = ix − x N − + w + N − ( x ) = ix − x N + w − N ( x ) ,w N ( x ) = ix − x N + w + N ( x ) , (19)where w ± j = O (1) at x → x j ±
0; and (iii) has the asymptotic behaviour w ( x ) = k + ˜ w −∞ ( x ) , ˜ w −∞ = O ( | x | − ) at x → −∞ ,w N ( x ) = − k + ˜ w ∞ ( x ) , ˜ w ∞ = O ( | x | − ) at x → ∞ . (20)Behaviour of w ( x ) is illustrated schematically in figure 1. We also notice that thecondition (19) defining the singularities is more restrictive than condition (10) consideredpreviously.In each interval I j we choose an arbitrary point X j ∈ I j , and define a function φ j ( x ) φ j ( x ) := ρ j exp " − i Z xX j w j ( ξ ) dξ , x ∈ I j , (21)where ρ j is a complex constant undefined, so far. One can readily verify that for theintroduced functions φ j ( x ) ( j = 1 , , . . . , N ) the following limits are validlim x → x j +0 φ j ( x ) = lim x → x j − φ j − ( x ) = 0 . (22)Indeed, for the left limits in points x j , we computelim x → x j − φ j − ( x ) = ρ j − lim x → x j − exp Z x j X j − (cid:20) ξ − x j − iw − j ( ξ ) (cid:21) dξ = 0 . universal form of complex potentials with spectral singularities x →−∞ [ φ ( x ) − e − ik x ρ − ] = lim x →∞ [ φ N ( x ) − e ik x ρ + ] = 0 , (23)where ρ ± are constants given by ρ − = ρ exp (cid:20) ik X − i Z −∞ X ˜ w −∞ ( ξ ) dξ (cid:21) , (24) ρ + = ρ N exp (cid:20) − ik X N − i Z ∞ X N ˜ w ∞ ( ξ ) dξ (cid:21) . (25)Note that the improper integrals in these formulas converge due to requirements (20).Let us now define the function ψ ( x ) := ( φ j ( x ) x ∈ I j x ∈ { x , x , . . . , x N } . (26)By the above properties of φ j ( x ) the so defined function ψ ( x ) is continuous on thewhole real axis and possesses asymptotic behaviour required for a SS-solution. Hence,for such a function to represent a SS-solution, it must be also continuously differentiableon the whole real axis. In the following theorem we prove that this goal can be achievedby the proper choice of the parameters ρ j . Theorem 3.1 (Sufficient condition) . Let X j ∈ I j for j = 0 , . . . , N and ρ be anarbitrary nonzero constant. Define amplitudes { ρ , . . . , ρ N } by the recurrent law asfollows ( j = 1 , , . . . , N ): ρ j = ρ j − X j − x j X j − − x j exp " i Z x j X j w + j ( ξ ) dξ − i Z x j X j − w − j ( ξ ) dξ . (27) Then for the function ψ ( x ) defined by (26) the following properties hold:(a) ψ ( x ) ∈ C ( R ) ;(b) there exist nonzero constants ρ ± such that lim x →±∞ [ ψ ( x ) − ρ ± e ± ik x ] = 0; (28) (c) for x ∈ I j , ψ ( x ) solves the equation − ψ ′′ + U j ( x ) ψ = k ψ (29) where U j ( x ) = − w j ( x ) − iw ′ j ( x ) + k . (30) Proof.
Bearing in mind already established properties of the functions φ j ( x ), for theproof of this theorem we only need to verify the continuity of the derivative ψ ′ in R .For x ∈ I this follows from the definition (26), because in each interval I j we compute ψ ′ ( x ) = − iw j ( x ) φ j ( x ) , x ∈ I j . (31) universal form of complex potentials with spectral singularities ψ ′ ( x ) is continuous in I j due to the continuity of w j ( x ) and φ j ( x ). Hence, it remainsto prove that ψ ′ ( x ) is continuous in the points x j , j = 1 , . . . , N . To this end, for eachpoint x j , we compute the left and right derivatives of ψ ( x ): ψ ′ ( x j −
0) = − lim ǫ → +0 ψ ( x j − ǫ ) ǫ = ρ j − X j − − x j exp " − i Z x j X j − w − j ( ξ ) dξ and ψ ′ ( x j + 0) = lim ǫ → +0 ψ ( x j + ǫ ) ǫ = ρ j X j − x j exp " − i Z x j X j w + j ( ξ ) dξ . Thus, if the amplitudes ρ j − and ρ j are connected through the relation (27), then thederivative of ψ ( x ) is continuous in R .
4. From a spectral singularity to exceptional point and to a bound state incontinuum
As we have demonstrated above, the complex potential U ( x ) of the form (12) alwayshas a SS solution, provided that the function w ( x ) satisfies certain conditions. Themost important of those conditions, which can be expressed by equation (9), requiresthe function w ( x ) to approach constant values ∓ k as x → ±∞ , respectively. In thoseconsiderations, it was important that the parameter k was real. Now we relax thisrequirement and consider physically relevant solutions with a complex k . In otherwords, k will be considered as a parameter that modifies the potential in order to obtaina prescribed solution. We note that a similar problem for PT − symmetric potentialswas recently addressed in [22] based on the fact that a SS can be viewed as a complexdiscrete energy corresponding to a solution with outgoing plane wave asymptotics.For the sake of simplicity, we assume that w ( x ) is a continuous function, whichallows to use solution (13) as a starting point. Generalization for discontinuous functions w ( x ) can be developed straightforwardly following the ideas of section 3; it is notpresented here.Consider a complex-valued continuous function w ( x ) which tends to ∓ k as x →±∞ , where k is an arbitrary complex constant. Then the function ψ ( x ), definedformally by equation (13), solves Schr¨odinger equation (1) with the potential givenby (12). Such a solution, however, may be irrelevant if it does not satisfy physicallymeaningful boundary conditions (i.e. conditions at x → ±∞ ). The asymptotic behaviorof w ( x ) implies that at large x the solution (13) behaves as ψ ∼ ρe ± ik x as x → ±∞ .Thus, solutions characterized by amplitudes growing with | x | , and hence physicallyirrelevant correspond to k in the lower complex half-plane, i.e., to Im k <
0. Suchsolutions will not be considered below. For other values of the complex k one candistinguish several cases (see the diagram in Fig. 2).If k is nonzero real (i.e., Im k = 0), then the solution (13) describes: • A spectral singularity universal form of complex potentials with spectral singularities Figure 2.
Diagram on the complex k -plane showing different features of the exactsolution (13). This is the case considered in the previous sections. For positive and negative k , thissolution is laser or a CPA, respectively.If k is in the upper complex half-plane (i.e., Im k >
0) then the solution (13)describes a bound state, i.e. satisfies the localization conditionlim x →∞ ψ ( x ) = lim x →−∞ ψ ( x ) = 0 . (32)Meantime, the nature of such a bound state can be different and depends on theassociated eigenvalue k : • A bound state occurs if Re k <
0, i.e., arg k ∈ ( π/ , π/ Z ∞−∞ ψ dx = 0.The respective bound state can be – stationary , if Re k = 0, – growing if Re k > – decaying if Re k < k outsideof continuum. • A bound state with the real part in continuum ( quasi-
BIC) takes place if the realpart of eigenvalue k is positive, i.e. lies in the continuous spectrum. In terms ofthe complex parameter k , this correspond to arg k ∈ (0 , π/ ∪ (3 π/ , π ). Forprevious discussion of solutions of these types see e.g. [23, 24]. • Exceptional point (EP) [25] is found if the quasi-self-orthogonality condition holds,i.e., Z ∞−∞ ψ dx = 0 [26]. An EP-solution can also be stationary, growing or decaying,what is determined by the real part of k as specified above. It also can be either abound state with a real part out of continuum or a quasi-BIC (i.e., having the realpart in the continuum.)Notice that the above classification of growing and decaying bound states stems fromthe physical interpretation of the stationary Schr¨odinger equation (1) as a reduced form universal form of complex potentials with spectral singularities − ∂ ψ∂x + U ( x ) ψ = − ∂ ψ∂t (33)after the ansatz ψ ∝ e − iωt , with ω = k and Re ω >
0, when growing with time solutioncorresponds to Im ω >
0; or alternatively as the optical parabolic approximation (orequivalently as time dependent Schr¨odinger equation with t replaced by dimensionlesspropagation distance z ) − ∂ ψ∂x + U ( x ) ψ = i ∂ψ∂z (34)after the ansatz ψ ∝ e ibz , where the propagation constant can be computed as b = − k .Respectively, bound states have Re b >
0, and growing (decaying) with z solutioncorresponds to Im b < b > w ( x ) is fixed and k is changing in the complex plane, one obtains a specific complex potential havingsolutions in a form of a SS or in a form of bound state with desired properties. Theonly exception of this rule is the parametric transition between a bound state and anEP-solution (where two bound states coalesce), because for such a transition to occurthe form of w ( x ) itself might need to be changed, and hence an additional free parametermay be needed. As a simple example illustrating this parametric dependence we consider k = i and the complex-valued function w ( x ) defined as w ( x ; z ) = − i tanh x + z sech x, (35)where z is an additional complex parameter. Using the explicit expression (13), with x = 0, we obtain the exact bound state solution ψ ( x ; z ) = exp (cid:26) iz π − x )) (cid:27) sech x. (36)Now it is straightforward to compute Z ∞−∞ ψ ( x ; z ) dx = 2 cos( πz )1 − z . (37)This integral becomes zero at z = z n = ± ( n + 1 /
2) and n = 1 , , . . . . Thus, ψ ( x ; z n )given by (36) corresponds to the EP of Eq. (1) with the potential U ( x ) generated by w ( x ; z n ). The resulting complex potential computed according to (12) reads U ( x ) = − ( z + 2)sech x + 3 iz tanh x sech x. (38)For real values of the parameter z the obtained potential is PT -symmetric, i.e. satisfiesthe property U ( x ) = U ∗ ( − x ) and belongs to the family of PT -symmetric Scarff IIpotentials [27, 28, 29]. universal form of complex potentials with spectral singularities
5. Self-dual spectral singularities for complex potentials (12) withreal-valued w ( x ) . Examples of SS-solutions. Let us now consider the obtained results on the existence of a SS solution and specificform of the potential in more conventional terms of the zeros of the transfer matrix. Forthis sake it will be convenient to rewrite the scattering problem (1) using the notationfor the Hamiltonian operator H : Hψ = k ψ, H = − ∂ xx + U ( x ) (39)and rewrite the Schr¨odinger equation (1) in the form of eigenvalue problem Hψ = k ψ .In this section, we consider only real valued functions w ( x ) which, as above, satisfythe boundary conditions w ( x ) → ∓ k as x → ±∞ . Respectively, the constant k isalso real in this section. In this case the Hamiltonian H admits an important additionalsymmetry which is expressed by the relation [30] ηH = H ∗ η, η = ∂ x + iw ( x ) , (40)which is close to the property of the pseudo-Hermiticity [31, 32, 33]. This implies thatfor any solution ψ with a real k one can construct another solution with the same k inthe form ( ηψ ) ∗ .For the scattering problem (39) one can introduce a pair of left (superscript “L”)and right (superscript “R”) Jost solutions which for real k are defined uniquely by theirasymptotics φ L1 ( x ; k ) → e ikx , φ L2 ( x ; k ) → e − ikx at x → −∞ , φ R1 ( x ; k ) → e ikx , φ R2 ( x ; k ) → e − ikx at x → + ∞ . (41)One can also introduce the 2 × M = M ( k ) which connects the leftand right Jost solutions as follows: φ L1 = M ( k ) φ R1 + M ( k ) φ R2 , φ L2 = M ( k ) φ R1 + M ( k ) φ R2 . (42)Any solution ψ ( x ) of (39) with real k is a linear combination of left and right Jostsolutions with some coefficients a L,R , b L,R : ψ ( x ) = a L φ L1 + b L φ L2 = a R φ R1 + b R φ R2 , (43)The relation among the coefficients is given by the transfer matrix a R b R ! = M M M M ! a L b L ! . (44)Thus a positive (negative) zero k ⋆ of matrix element M corresponds to a spectralsingularity describing the laser (CPA) solution. Conversely, a positive (negative) zero k ⋆ of M describes a CPA (laser).While the definition of transfer matrix introduced above is valid for any scatteringpotential U ( x ), the peculiar property (40) imposes additional relations among the universal form of complex potentials with spectral singularities η operator each Jost solution in(41). Using that for large x the action of η can be approximated by lim x →±∞ η = ∂ x ∓ ik and also using the fact that the Jost solutions with real k are defined uniquely by theirasymptotic behaviours, we deduce the following relations between the Jost solutions:( ηφ L1 ) ∗ = − i ( k + k ) φ L2 , ( ηφ R1 ) ∗ = − i ( k − k ) φ R2 , ( ηφ L2 ) ∗ = i ( k − k ) φ L1 , ( ηφ R2 ) ∗ = i ( k + k ) φ R1 . (45)Using these relations together with (42), we obtain that for all real k the transfer matrixelements are connected as M ( k ) = M ∗ ( k ) k + k k − k , M ( k ) = M ∗ ( k ) k − k k + k ,M ( k ) = − M ∗ ( k ) , M ( k ) = − M ∗ ( k ) . (46)To be specific, below we consider the case k >
0. Then we readily concludethat M ( k ) = 0, which corresponds to a laser emitting at wavenumber k = k .This laser solution obviously recovers the already known exact solution ψ ( x ) givenby the explicit expression (13). If at some k ⋆ the two conditions M ( k ⋆ ) = 0 and M ( k ⋆ ) = 0 are verified simultaneously. This situation is typical for, although notlimited to, PT − symmetric systems [11] and such SSs at k ⋆ are sometimes referredto as self-dual [34]. Now one can verify the spectral singularity at k determined bythe solution (13) is generically non -self-dual, i.e. in a general case M ( k ) is nonzero.Indeed, evaluating M ( k ) according to L’Hˆopital’s rule, we obtain M ( k ) = 2 k dM ∗ ( k ) dk . (47)Thus if the spectral singularity k is of the first order, i.e., if M ( k ) = 0 and dM ( k ) /dk = 0, then M ( k ) = 0 and the SS k is non-self-dual. This leads us to thefollowing necessary and sufficient condition: the SS k corresponding to solution (13) isself-dual, iff it is a zero of the second or higher order of one of diagonal elements of thetransfer matrix, i.e., iff the condition one of the conditions M ( k ) = dM ( k ) /dk = 0 or M ( k ) = dM ( k ) /dk = 0 is verified. The latter observation is particularly curious in view of the fact that if there isa SS k ⋆ different from ± k , it is always self-dual. Indeed it follows from (46) that if M ( k ⋆ ) = 0 with k ⋆ = ± k , then M ( k ⋆ ) = 0, as well. Thus, for the class of complexpotentials under consideration an interesting situation is possible when the potentialfeatures an ordinary (non-self-dual) SS k corresponding to the exact solution (13) and,at the same time, has a self-dual SS at k ⋆ = k .A representative feature of self-dual SSs for potentials (12) with real w ( x ) is thatthe amplitudes of the corresponding CPA and laser solutions, denoted below by ψ las ( x )and ψ CPA ( x ), respectively, are connected by a simple algebraic relation. Indeed, let k ⋆ = ± k be a self-dual SS. Without loss of generality we can normalize those solutions universal form of complex potentials with spectral singularities | ψ las ( x ) | → ρ ⋆ and | ψ CPA ( x ) | → ρ ⋆ as x → ±∞ , where ρ ⋆ is a positiveconstant. Then one can show that the laser and CPA amplitudes are related as | ψ CPA | + | ψ las | + k k ⋆ ( | ψ las | − | ψ CPA | ) = 2 ρ ⋆ . (48)In order to prove this identity, let us again return from the scattering problem in theform (39), to the Schr¨odinger equation (1) with potential (12). It is straightforward tocheck that any solution ψ ( x ) of equation (1) with real k satisfies the identity (“theconservation law”) [35] | ηψ | + ( k − k ) | ψ | = const . (49)Let ψ in (49) be the laser solution ψ las ( x ) and k in (49) be the corresponding wavenumber k ⋆ . Due to the symmetry (40) a new function φ ( x ) = ( ηψ las ) ∗ is also a solution ofSchr¨odinger equation (1) with the same k ⋆ . Asymptotic behavior of φ ( x ) indicates thatthe latter is a CPA solution, and therefore it is proportional to the normalized CPAsolution ψ CPA ( x ). Comparing the amplitudes of both solutions, we obtain | ψ CPA | = | ηψ las | ( k ⋆ − k ) . (50)Combining (49) and (50), we obtain (48). Now we present several numerical examples illustrating the discussed self-dual SSs. Tothis end we consider w ( x ) to be an odd real-valued function which satisfies the boundaryconditions (9). The respective potential U ( x ) in the form (12) is an even function (i.e., itis not PT symmetric). It is clear that if w ( x ) ia a monotonous function then no self-dualSSs can exist, because in this case the derivative w ′ ( x ) is sign-definite, meaning thatthere is either only gain (then CPA is impossible) or only loss (then laser is impossible).Therefore, in order to have a self-dual SS and the corresponding CPA-laser, one needsto employ more sophisticated nonmonotonic functions w ( x ).As a representative example, here we consider w ( x ) = − k erf ( x ) (cid:16) − γe − x (cid:17) , k = 1 , (51)where erf ( x ) is the error function and γ is a real parameter that tunes the shape offunction w ( x ). For γ = 0, the potential U ( x ) generated according to (12) correspondsto amplifying media on the whole space. However, for sufficiently large positive γ , theresulting potential U ( x ) corresponds to spatially localized absorbing region sandwichedbetween two amplifying domains [see Fig. 4(a,b) below for representative plots of theresulting complex potential]. It is interesting to notice that the quantityΓ = w ( −∞ ) − w ( ∞ ) = Im Z ∞−∞ U ( x ) dx (52)describes the balance of the total energy either pumped into, Γ >
0, or absorbed byΓ <
0, the system. In our case Γ = 2 k , i.e., it is fixed by k , and thus does not dependon the value γ controlling distribution of the energy in space. universal form of complex potentials with spectral singularities Figure 3. (a) Logarithmic plots of the absolute values | M ( k ) | (bold blue curve)and | M ( k ) | (thin red curve) for the potential U ( x ) generated by function (51) with γ = γ (1) ⋆ ≈ . w ( x ) and its derivative, as well as laser and CPAsolutions corresponding to the self-dual spectral singularity are shown in Fig. 4(a-f)below. (b) Logarithmic plots of magnitudes of reflection | R L | = | R R | (bold magentacurve) and transmission | T | (thin green curve) coefficients computed from the transfermatrix elements plotted in (a). In contrast to the exact laser solution ψ ( x ) given by (13) and existing for any γ at k = k , eventual self-dual SSs can be found only for isolated values of γ . Obtaininga self-dual SS is reduced to numerical solution of a system of two equationsRe M ( k ) = 0 , Im M ( k ) = 0 , k = k , (53)with respect to two real variables: γ and k .Computing numerically the Jost solutions and the elements of the transfer matrix,we tune the free parameter γ and wavevector k in order to reach a self-dual SS. Thesmallest positive value of γ that enables a self-dual SS is γ = γ (1) ⋆ ≈ . γ (1) ⋆ are shown in figure 3(a), where twospectral singularities at different values of k are well visible. The first spectral singularitycorresponds to the zero of M ( k ) at k = k = 1 and, equivalently, to the zero of M ( k )at k = − k = − ψ given by (13). Notice thatanother diagonal element is not zero for this spectral singularity: M ( k ) = 0 and universal form of complex potentials with spectral singularities Figure 4.
Upper row: plots of w ( x ) and w ( x ) , w ′ ( x ) given by (51) with γ = γ (1) ⋆ ≈ . γ (1) ⋆ and k (1) ⋆ . Two bottom rows showthe coexisting laser and CPA solutions for the next self-dual SS at γ (2) ⋆ and k (2) ⋆ . M ( − k ) = 0. Thus this spectral singularity is not self-dual. The second spectralsingularity occurs at k (1) ⋆ ≈ ± .
490 and is self-dual, since both diagonal elements arezero at this wavevector: M ( ± k (1) ⋆ ) = M ( ± k (1) ⋆ ) = 0.CPA and laser solutions coexisting at the self-dual SS are shown in Fig. 4(c,d,e,f).The laser solution is an odd function of x (hence its phase undergoes a π jump at x = 0), whereas the CPA solution is an even function of x and has the intensitypeak at x = 0. Amplitudes of the coexisting laser and CPA solutions shown in universal form of complex potentials with spectral singularities ρ ⋆ = 1. Interestingly, the amplitude of the CPA solution in the central region is largerthan the amplitude of background radiation. This highlights the fact that enhancedabsorption results from the constructive interference of coherent scattered waves in thecentral, i.e. absorbing domain (notice that similar type of behavior was observed in therecent experiments [36]). Respectively, the laser solution corresponds to the destructiveinterference in the central region where the solution amplitude has a node.Further, we use transfer matrix in order to evaluate the transmission coefficient T ( k ) and left and right reflection coefficients, R L ( k ) and R R ( k ), using the standardformulas T = 1 M , R L = − M M , R R = M M . (54)In the case at hand the left and right transmission coefficients coincide [9], i.e. T := T L = T R . The amplitudes of obtained scattering coefficients are plotted infigure 3(b); notice that in view of relations (46) | R L ( k ) | ≡ | R R ( k ) | , i.e. the left and rightreflection coefficients differ only by phases. The difference between the two coexistingSSs becomes evident if one compares the behaviour of the scattering coefficients forpositive and negative values of k . For the self-dual SS, all three scattering coefficientsdiverge at k (1) ⋆ and − k (1) ⋆ producing two peaks of infinite height. The ordinary (non-self-dual) SS at k = ± k produces a single peak at k = k without a twin in the negative k -half-axis. Additionally, in figure we observe that the potential becomes bidirectionallyreflectionless at k ≈ ± .
116 where both reflection coefficients vanish simultaneously.New self-dual SSs can be found for larger values of the parameter γ in (51). Forinstance, upon increasing γ the next spectral singularity occurs at γ (2) ⋆ ≈ .
082 and k (2) ⋆ ≈ . x , respectively. Their shapes are illustrated in figure 4(g,h,i,j). Furtherincrease of γ leads to the next spectral singularity at γ (3) ⋆ ≈ .
203 and k (3) ⋆ ≈ . k at anyvalue of γ , the potential (51) features a sequence of self-dual SSs with γ (1) ⋆ < γ (2) ⋆ < . . . and k < k (1) ⋆ < k (2) ⋆ . . . . Notice that in the presented example, the energy balanceintegral Γ remains constant, i.e. when increasing gain one also has to increase losses.We have also computed numerically the entire spectrum of eigenvalues of operator H and observed that the increase of γ above each self-dual SS leads to the bifurcationof a new complex conjugate pair of eigenvalues from the continuous spectrum. Thespectrum is purely real for 0 ≤ γ ≤ γ (1) ⋆ , but features a single complex conjugatepair for γ (1) ⋆ < γ ≤ γ (2) ⋆ , two complex conjugate pairs for γ (2) ⋆ < γ ≤ γ (3) ⋆ , etc. Thisis illustrated in Fig. 5, where the entire spectra of eigenvalues are illustrated for twovalues of γ . Thus the increase of the parameter γ above the first CPA-laser threshold γ (1) ⋆ triggers the phase transition from purely real to complex spectrum (such phasetransition was discovered numerically in [37] and described analytically in [38]). universal form of complex potentials with spectral singularities Figure 5.
Spectrum of eigenvalues for the Schr¨odinger operator with the potential(51) for (a) γ = 4 and (b) γ = 11 .
5. Thick line shows the continuous spectrum andcircles indicate the isolated eigenvalues. For γ ≤ γ (1) ⋆ ≈ .
468 the spectrum is purelyreal and continuous (not shown in the figure).
6. Discussion and conclusion
The main result of this work is that in a quite general physical situation the existence ofa SS in the spectrum of a one-dimensional Sch¨odinger operator implies a universalrepresentation of the complex potential, which is given by (12). This form of thepotential, being a subject of many theoretical studies in the last decade, is not onlya formal algebraic construction but models experimentally feasible potentials, sayrealized recently in acoustic systems [36]. Complex potentials of this form can be alsoimplemented in a coherent atomic system driven by laser fields [40].Functional dependence of the potential (12) on only one base function w ( x ) andon the wavevector k at which a SS is observed, allows to engineer complex potentialsfeaturing SSs at a given wavelength.We have shown that the corresponding eigenvalue problem has an exact solution(13) which can be either a CPA or laser, both corresponding to real k . Relaxing thecondition of the reality of k , i.e. considering it in the complex plane, by changing k one can transform the complex potential such that instead of a SS its spectrum cancontain bound states, quasi-bound states in continuum, and exceptional points.Generically a SS singularity described by the exact solution (13) is simple andnon-self-dual. For the particular case when the base function w ( x ) is real-valued weestablished that for the exact SS-solution to be a self-dual SS it must be also a secondorder SS. A numerical example of a potential featuring one simple SS (correspondingto the exact solution) and a set of self-dual SSs was considered in details. Wehave additionally computed the spectrum of eigenvalues of the corresponding complexpotentials and confirmed that a transition through a self-dual spectral singularitygenerically leads to a bifurcation of a complex-conjugate pair of discrete eigenvaluesfrom an interior point of the continuous spectrum.As concluding remarks, we mention two straightforward generalizations of thepresented theory. First, one can consider base functions w ( x ) that approach differentvalues at the infinities: lim x →±∞ w ( x ) = k ± . In particular, one can construct “one-sided” spectral singularities, i.e. a “hybrid” between SS at one infinite and bound stateat another infinity. However, in this case the potential U ( x ) is not localized, unless universal form of complex potentials with spectral singularities k +0 = ± k − .Second, for real-valued functions w ( x ) the exact solution (13) can bestraightforwardly generalized on the nonlinear case. Indeed, incorporating in our modelthe cubic (Kerr) nonlinearity, instead of (39) we obtain the nonlinear eigenvalue problem − ψ xx + ( − w − iw x + k ) ψ + σ | ψ | ψ = k ψ. (55)General properties of this equation were discussed in [35] and solution of the type (13),although not a SS-solution, was addressed in [39]. The exact SS-solution ψ ( x ) of (55)can be found in the same form as in Eq. (13), with the simple shift of the nonlineareigenvalue: k = k + σ | ρ | . Since the found nonlinear solution features purely outgoingor purely incoming (depending on the sign of k ) wave boundary conditions, it paves theway towards the implementation of a CPA for nonlinear waves which was experimentallyimplemented in a Bose-Einstein condensate [41] and theoretically predicted for arraysof optical waveguides [42]. Acknowledgments
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