Construction of non-PT-symmetric complex potentials with all-real spectra
aa r X i v : . [ m a t h - ph ] D ec Construction of non-
P T -symmetriccomplex potentials with all-real spectra
Jianke Yang ∗ Department of Mathematics and StatisticsUniversity of VermontBurlington, Vermont, USA
Abstract:
We review recent work on the generalization of PT symmetry. We show that,in addition to PT -symmetric complex potentials, there are also large classesof non- PT -symmetric complex potentials which also feature all-real spectra.In addition, some classes of these non- PT -symmetric potentials allow phasetransitions which do or do not go through exceptional points. These non- PT -symmetric potentials are constructed by a variety of methods, such as the sym-metry and supersymmetry methods and the soliton theory. A generalization of PT symmetry in multi-dimensions is also reviewed. Linear paraxial propagation of light in an optical waveguide is governed by theSchr¨odinger equation [1, 2, 3]iΨ z + Ψ xx + V ( x )Ψ = 0 , (1)where z is the distance of propagation, x is the transverse coordinate, V ( x ) is acomplex potential whose real part is the index of refraction and the imaginarypart represents gain and loss in the waveguide. This same equation also arises innon-Hermitian quantum mechanics [4, 5] and Bose-Einstein condensates [6], inwhich case z is the time variable. Looking for eigenmodes of the form Ψ( x, z ) =e i µz ψ ( x ) we arrive at the eigenvalue problem Lψ = µψ, (2)where L = ∂ xx + V ( x ) is a Schr¨odinger operator, and µ is an eigenvalue.All-real spectrum for this Schr¨odinger operator with a complex potential isa sought-after property not only in non-Hermitian quantum mechanics, but alsoin optics and Bose-Einstein condensates. In quantum mechanics, µ is the energy ∗ Email: [email protected] L isparity-time ( PT ) symmetric, i.e., it is invariant under the joint transformationsof x → − x and complex conjugation, then its spectrum can be all-real. This PT symmetry is equivalent to the condition on the complex potential that V ∗ ( x ) = V ( − x ) , (3)where the asterisk * represents complex conjugation. In optics, this conditionmeans that the refractive index needs to be an even function in space, and thegain-loss profile needs to be an odd function in space. A simple reason for thisall-real spectrum of a PT -symmetric potential is that its eigenvalues alwaysappear in complex conjugate pairs. This is because for such potentials, if µ isan eigenvalue with eigenfunction ψ ( x ), then by taking the complex conjugateof Eq. (2) and switching x → − x , we see that µ ∗ would also be an eigenvaluewith eigenfunction ψ ∗ ( − x ). This eigenvalue symmetry restricts the appearanceof complex eigenvalues and facilitates the realization of an all-real spectrum.But this PT symmetry does not necessarily guarantee an all-real spectrum, andphase transition can occur when conjugate pairs of complex eigenvalues appearin the spectrum [2, 4, 7]. PT symmetry has found many optical applications, such as unidirectionalreflectionless metamaterials [8], PT lasers [9, 10], and non-reciprocity in PT -symmetric whispering-gallery microcavities [11]. In these optical applications,the refractive index and gain-loss profiles of the waveguide were carefully de-signed so as to respect PT symmetry. In a PT setting, the gain-loss profilemust be anti-symmetric, which could be restrictive. The pursuit of non- PT -symmetric potentials with more flexible gain-loss profiles and all-real spectrais thus an interesting question. In recent years, various techniques have beendeveloped to construct non- PT -symmetric potentials with all-real spectra, andthey will be reviewed in this article (a brief review on some of these results couldalso be found in [12]). P T -symmetric potentials with all-real spec-tra and exceptional-point-mediated phase tran-sition
To derive non- PT -symmetric complex potentials with all-real spectra, one strat-egy is to impose an operator symmetry in order to guarantee conjugate-paireigenvalue symmetry [13]. Like the case of PT -symmetric potentials, thisconjugate-pair eigenvalue symmetry guarantees that either the spectrum of L isall-real, or a phase transition occurs when pairs of complex eigenvalues appear.2o execute this strategy, we observe that if there exists an operator η suchthat L and its complex conjugate L ∗ are related by a similarity relation ηL = L ∗ η, (4)then the eigenvalues of L would come in conjugate pairs if the kernel of η is empty[13]. This operator relation resembles the condition for pseudo-Hermiticity [5],but we do not require η to be invertible here.If we let η = P , where P is the parity operator x
7→ − x , then this η operatorhas an empty kernel, and the similarity condition (4) yields V ( − x ) = V ∗ ( x ),which recovers the well-known class of PT -symmetric potentials. However,when branching out to different choices of η , a completely real spectrum canbe obtained for an arbitrary choice of the gain-loss distribution by a judiciousconstruction of the index of refraction. This will be demonstrated below where η is chosen as a differential operator. First, we consider the simplest choice of a differential η operator, η = ∂ x + a ( x ) . (5)Substituting this η and operator L into the similarity condition (4), we get thefollowing two equations a x = i Im( V ) , a xx − V x = ( a ) x . (6)The second equation can be integrated once, and we get a x − V = a + ξ , (7)where ξ is a constant. Utilizing (6), this equation becomesRe( V ) = − a − ξ . (8)Eqs. (6)-(8) show that a ( x ) is a purely imaginary function, and ξ is a realconstant. Denoting a ( x ) = i g ( x ), where g ( x ) is an arbitrary real function, weget Re( V ) = g ( x ) − ξ and Im( V ) = g ′ ( x ), with the prime representing thederivative. The constant ξ can be eliminated by a gauge transformation toEq. (1), and thus the resulting complex potential is V ( x ) = g ( x ) + i g ′ ( x ) . (9)These potentials were called type-I potentials in [13]. They generalized thepotentials of the same form in [14, 15], where special choices of the g ( x ) functionwere taken (see sections 2.3 and 5 for reviews).Compared to PT -symmetric potentials, a distinctive feature of these type-I potentials is that the gain-loss profile g ′ ( x ) is now arbitrary since g ( x ) is3rbitrary. But due to the symmetry relation (4), the spectra of these potentialscan still be all-real, just like PT -symmetric potentials. This possibility for all-real spectra for arbitrary gain-loss profiles is made possible by a judicious choiceof refractive indices in relation to the gain-loss profiles.As an example, we take g ( x ) = tanh[2( x + 2 . − tanh( x − .
5) + c , (10)where c is a real constant. In Fig. 1, we show two potentials of the form(9), with the c value taken as 0 and − . c = 0 has a completely real spectrum, and increasing c willmaintain the reality of the spectrum as more discrete eigenvalues bifurcate offthe edge of the continuous spectrum. However, as c is decreased, the spectrumwill undergo a phase transition at c ≈ − . µ ≈ . c = − .
3. We stress that this phase transition is induced by going throughan exceptional point, which is a common scenario for phase transition [4, 7]. −10 0 10−2024
Potential x V −2 0 2 4−0.200.2 Spectrum
Re[ µ ] I m [ µ ] −10 0 10−2024 Potential x V −2 0 2 4−0.200.2 Spectrum
Re[ µ ] I m [ µ ] Figure 1: Spectra of type-I potentials (9) with g ( x ) given in (10). Upper row: c = 0; lower row: c = − .
3. In the potentials, the solid blue line is Re( V ),and the dashed pink line is Im( V ). Adapted from [13].4 .2 Type-II potentials Type-I potentials (9) come from taking the simplest form of a differential η operator [i.e., a first-order operator (5)]. By increasing the order of this differ-ential operator, more families of potentials arise. Let η now be a second-orderoperator, η = ∂ xx + a ( x ) ∂ x + b ( x ) . (11)Inserting this η into the similarity condition (4) and collecting coefficients of thesame order of derivatives on the two sides of this condition, we get ηL L ∗ η∂ x ∂ x a a∂ x V V ∗ + 2 a ′ ∂ x V a + 2 V ′ V ∗ a + a ′′ + 2 b ′ ∂ x V b + V ′ a + V ′′ V ∗ b + b ′′ Setting these coefficients in ηL and L ∗ η to match each other, we get a systemof equations which can be solved from top to bottom. From the ∂ x coefficients,we get a ′ ( x ) = i Im( V ). Setting a ( x ) = i g ( x ), where g ( x ) is a real function,we obtain Im( V ) = g ′ ( x ). Inserting this a ( x ) formula into the ∂ x equation andintegrating once, we get b = Re( V ) − g + i2 g ′ + c , where c is a constant.Now we insert these a ( x ) and b ( x ) solutions into the ∂ x equation. Aftersimple algebra, this equation becomes[Re( V ) g ] ′ = g g ′ − g ′′′ g − c g ′ g, from which we can solve the refractive index Re( V ) asRe( V ) = 14 g + g ′ − g ′′ g + c g − c , where c is a real constant. The overall constant c can be removed withoutloss of generality.Putting the above results together, we get potentials V ( x ) = 14 g + g ′ − g ′′ g + c g + i g ′ , (12)where g ( x ) is an arbitrary real function, and c is an arbitrary real constant.These potentials were called type-II potentials in [13], and they generalizedpotentials of the same form but with c ≤ g ′ ( x ) of these type-II potentialsis also arbitrary, but their spectra can still be all-real due to the symmetryrelation (4).As an example, we take the same function g ( x ) as in (10). The spectrum ofthis potential with c = 1 and c = − c and decreasing c will maintain the all-real spectrum. If c is increasedabove a certain threshold (which is approximately 2.535), a phase transitionoccurs, where a pair of real eigenvalues coalesce and form an exceptional point,which then bifurcates off the real axis and creates a pair of complex eigenvaluesafterwards. This can be seen in the lower row of Fig. 2 for c = 4. In thislower row, an overall real constant ( c + c /c ) / −10 0 10−2024 Potential x V −2 0 2 4−0.200.2 Spectrum
Re[ µ ] I m [ µ ] −10 0 10−2024 Potential x V −2 0 2 4−0.200.2 Spectrum
Re[ µ ] I m [ µ ] Figure 2: Spectra of type-II potentials (12) with g ( x ) given in (10). Upper row: c = 1 and c = −
1; lower row: c = 1 and c = 4. Adapted from [13]. For the two types of potentials (9) and (12), an exceptional-point-mediatedphase transition is in general possible (see Figs. 1 and 2). But under certainrestrictions on these potentials, all-real spectra can be guaranteed.For type-I potentials (9), it was shown by Tsoy et al. [14] that, if g ( x ) isa single-humped localized real function, then its spectrum is strictly real. This6esult was based on an observation by Wadati [15] that the Zakharov-Shabat(ZS) spectral problem [17] v x + i ζv = g ( x ) v , v x − i ζv = − g ( x ) v , (13)with ζ being a spectral parameter, can be transformed into the Schr¨odingereigenvalue problems ψ xx + V ( x, t ) ψ = µψ, φ xx + V ∗ ( x, t ) φ = µφ, with V ( x ) being the type-I potential (9) and µ = − ζ , through the transforma-tion ψ = v − i v , φ = v + i v . (14)This means that, in order for the type-I potential (9) to have all-real µ spec-trum, the necessary and sufficient condition is that the ζ spectrum of the ZSspectral problem (13) is either real or purely imaginary (note that the contin-uous spectrum of the ZS problem is the real ζ axis). It was shown by Klausand Shaw [18] that when g ( x ) is a single-humped localized real function, thenall discrete eigenvalues of the ZS problem are purely imaginary, and thus type-Ipotentials (9) have all-real spectra.For type-II potentials (12), it was shown by Andrianov et al. [16] that if c ≤
0, then the spectrum is strictly real. The proof is based on supersymmetry(see Sec. 4). Specifically, when c = − ǫ ≤
0, with ǫ being a real parameter,then we have the following intertwining operator relation,[ − ∂ x + W ( x )] [ ∂ xx + V ( x )] = [ ∂ xx + V ( x )] [ − ∂ x + W ( x )] , (15)where V ( x ) is the complex type-II potential (12), W ( x ) = g ′ + ǫ g −
12 i g, and V is a real potential, V ( x ) = 14 g + 2 gg ′′ − g ′ − ǫg ′ − ǫ g . The intertwining relation (15) shows that the Schr¨odinger operators ∂ xx + V ( x )and ∂ xx + V ( x ) are related by a similarity transformation, and thus they sharethe same spectrum. Since the spectrum of the real potential V is all-real, thespectrum of the type-II potential (12) is then all-real as well. Note that for c >
0, such an intertwining operator relation does not exist, and the supersymmetryapproach does not apply. In such cases, phase transition can occur in type-IIpotentials as Fig. 2 shows. 7
Non-
P T -symmetric potentials with all-real spec-tra and exceptional-point-free phase transi-tion
Extending the symmetry approach of the previous section, additional new typesof complex potentials with all-real spectra can be constructed [19]. More inter-estingly, these potentials exhibit exceptional-point-free phase transition, whichis very novel.In this construction, instead of choosing η in Eq. (4) as pure differentialoperators, we now take η to be a combination of the parity operator P anddifferential operators. In the simplest case, we take η to be a combination ofthe parity operator and a first-order differential operator, i.e., η = P [ ∂ x + h ( x )] , (16)where h ( x ) is a complex function to be determined. Substituting this η into thesimilarity condition (4), we get the following two equations V ( x ) − V ∗ ( − x ) = 2 h ′ ( x ) , (17)[ V ( x ) − V ∗ ( − x )] h ( x ) = h ′′ ( x ) − V ′ ( x ) . (18)From the first equation, we see that [ h ∗ ( − x )] x = h ′ ( x ); thus h ∗ ( − x ) = h ( x ) + c , (19)where c is a constant. Substituting Eq. (17) into (18) and integrating once,we get V ( x ) = h ′ ( x ) − h ( x ) + c , (20)where c is another constant. Lastly, inserting (19) and (20) into Eq. (17), weobtain c + 2 c h ( x ) + c − c ∗ = 0 . (21)In order for the potential V ( x ) in (20) not to be a constant, the function h ( x )should not be identically zero. Thus, Eq. (21) dictates that c = 0 and c is real.The former condition means that the complex function h ( x ) is PT -symmetricin view of Eq. (19). Regarding the latter condition, since a real constant ina potential can be easily removed by a simple shift of the eigenvalue, we canset c = 0 without loss of generality. In the end, we find that for new complexpotentials of the form V ( x ) = h ′ ( x ) − h ( x ) , (22)where h ( x ) is a PT -symmetric complex function, i.e., h ∗ ( x ) = h ( − x ), theSchr¨odinger operator L satisfies the similarity condition (4) with η given in(16). Because of this, the eigenvalues of L exhibit complex-conjugate symme-try. Hence, the spectrum of L can be all-real, but phase transition may alsooccur, similar to PT -symmetric potentials as well as non- PT -symmetric poten-tials of the previous section. 8n these new potentials, h ( x ) is an arbitrary PT -symmetric function. Be-cause of this, simple algebra shows that these potentials can accommodate anyarbitrary gain-loss profile [19], analogous to type-I and type-II potentials of theprevious section.A peculiar property of this new class of non- PT -symmetric potentials isthat, if these potentials are localized, then they will not admit any discrete realeigenvalues. This contrasts the previous non- PT -symmetric potentials (9) and(12), where discrete real eigenvalues are very common (see Figs. 1 and 2).To prove this statement, we recall that for any localized potential, the con-tinuous spectrum of the Schr¨odinger operator L is the semi-infinite interval −∞ ≤ µ ≤
0; and discrete real eigenvalues, if any, are positive numbers. Sup-pose µ = k , with k >
0, is a discrete real eigenvalue in the localized potential(22). Since L is a second-order differential operator, its discrete eigenvalue µ can only have geometric multiplicity one, meaning that the corresponding eigen-function ψ is unique (up to a constant multiple). Applying the operator η tothe eigenvalue equation Lψ = k ψ and recalling the symmetry relation (4), weget L ∗ ( ηψ ) = k ( ηψ ). Taking the complex conjugate of this equation, we get L ( ηψ ) ∗ = k ( ηψ ) ∗ . This means that ( ηψ ) ∗ is also an eigenfunction of L atthe same eigenvalue µ . Thus, ( ηψ ) ∗ and ψ must be linearly dependent on eachother, i.e., ( ηψ ) ∗ = αψ , where α is a complex constant. In view of the expressionof η in Eq. (16), this relation can be rewritten as[ ∂ x + h ( x )] ψ ( x ) = α ∗ ψ ∗ ( − x ) . (23)Now we examine this relation as x → ±∞ . Since the potential V ( x ) is localized, h ( x ) is localized as well. From the eigenvalue equation (2), we see that the large- x asymptotics of ψ ( x ) is ψ ( x ) → a ± e − k | x | , x → ±∞ , where a ± are complex constants which cannot be both zero. Since h ( x ) is local-ized, as x → ±∞ , the contribution of the h ( x ) term in Eq. (23) is subdominantand will be ignored. Then, inserting the above ψ -asymptotics into (23), we gettwo parameter conditions − ka + = α ∗ a ∗− , ka − = α ∗ a ∗ + . Dividing these two equations and rearranging terms, we get | a + | + | a − | = 0 , which is impossible since a ± cannot be both zero. Thus, localized potentials(22) do not admit discrete real eigenvalues.The fact of localized potentials (22) not admitting discrete real eigenvaluesis a distinctive property, and it has important implications. Since there are nodiscrete real eigenvalues, a phase transition in these potentials clearly cannotbe induced from collisions of such eigenvalues through an exceptional point.Instead, complex eigenvalues will have to bifurcate out from the continuous9pectrum. Below, we will use an example to show that this is exactly the case.In this example, we take h ( x ) = d sech x + i d sech x tanh x, (24)which is PT -symmetric for real constants d and d . We also fix d = 1. Thenfor two different d values of 1 and 2, the resulting non- PT -symmetric localizedpotentials and their spectra are plotted in Fig. 3. We see that neither spec-trum contains discrete real eigenvalues, which corroborates our analytical resultabove. When d = 1, the spectrum is all-real (see the upper right panel). Butwhen d = 2, a conjugate pair of discrete eigenvalues µ ≈ − . ± . d ≈ . µ ≈ − . µ , rather than from a single coalesced eigenfunction. Thisreveals two facts: (1) these discrete eigenmodes bifurcate out from continuouseigenmodes, rather than embedded isolated eigenmodes, in the interior of thecontinuous spectrum; (2) this phase transition does not go through an excep-tional point. The second fact is particularly significant, because all phase tran-sitions reported before in both finite- and infinite-dimensional non-Hermitiansystems occurred either due to a collision of real eigenvalues forming an excep-tional point, where different eigenvectors or eigenfunctions coalesce [4, 7, 13],or through an exotic singular scenario, where complex eigenvalues bifurcate outfrom infinity of the real axis [20]. This is the first instance where a phase tran-sition occurs without an exceptional point or a singular point. Very recently,an analytical explanation of this mysterious bifurcation was given by Konotopand Zezyulin [21] through the splitting of self-dual spectral singularity. P T -symmetric potentialswith all-real spectra using supersymmetry
The concept of supersymmetry (SUSY) was first introduced in quantum fieldtheories and high-energy physics (see [22] and the references therein). Subse-quently, SUSY was utilized in quantum mechanics to construct analyticallysolvable potentials. This construction is based on the factorization of theSchr¨odinger operator into the product of two first-order operators. Switchingthe order of these two first-order operators gives another Schr¨odinger opera-tor with a new potential (called the partner potential ) which shares the samespectrum as the original potential (except possibly a single discrete eigenvalue).SUSY can establish perfect phase matching between modes in the partner po-tentials, which has motivated applications such as mode converters in SUSYoptical structures [23]. Extending the idea of SUSY, parametric families ofcomplex potentials with all-real spectra can be constructed [16, 24, 25, 26, 27].10 x V d = 1 −5 0 5−101 Re( µ ) I m ( µ ) −5 0 5−2−10123 x V d = 2 −2 0 2−101 Re( µ ) I m ( µ ) Figure 3: Spectra of localized potentials (22) with h ( x ) given in (24) and d = 1(the d values are shown inside the panels). Left column: real (solid blue) andimaginary (dashed red) parts of the complex potentials. Right column: spectraof potentials in the left column (the red arrows in the lower panel indicate thatthe two complex eigenvalues in the spectrum bifurcate out from the red dotin the interior of the continuous spectrum when a phase transition happens).Adapted from [19].Let us employ the idea of SUSY to construct complex potentials with all-realspectra, following [24, 27, 28].Suppose V ( x ) is a potential with all-real spectrum, and µ (1) is an eigenvalueof this potential with eigenfunction ψ (1) , i.e., (cid:20) d dx + V ( x ) − µ (1) (cid:21) ψ (1) = 0 . (25)We first factorize the linear operator in this equation as − d dx − V ( x ) + µ (1) = (cid:20) − ddx + W ( x ) (cid:21) (cid:20) ddx + W ( x ) (cid:21) . (26)The function W ( x ) in this factorization can be obtained by requiring ψ (1) toannihilate d/dx + W ( x ), and this gives W ( x ) as W ( x ) = − ddx ln( ψ (1) ) . (27)11t is easy to directly verify that this W ( x ) does satisfy the factorization equa-tion (26).Now we switch the two operators on the right side of the above factorization,and this leads to a new potential V ( x ), − d dx − V ( x ) + µ (1) = (cid:20) ddx + W ( x ) (cid:21) (cid:20) − ddx + W ( x ) (cid:21) , (28)where V = V − W x . (29)This V potential is referred to as the partner potential of V . It is known thatfor any two operators A and B , AB and BA share the same spectrum in general(except for a possible difference in the zero eigenvalue when the kernel of A or B is non-empty). Then, in view of the two factorizations (26) and (28), we seethat the spectrum of V is that of V , but with µ (1) generically removed.The new potential V , however, is only real or PT -symmetric if V is so. Inorder to derive non- PT -symmetric potentials, we build a new factorization forthe V potential, − d dx − V ( x ) + µ (1) = (cid:20) ddx + f W ( x ) (cid:21) (cid:20) − ddx + f W ( x ) (cid:21) . (30)The function f W in this new factorization can be derived as follows [24, 27].Equating this new factorization with the previous one in (28), we get f W x + f W = W x + W . Denoting f W = W + φ , we see φ satisfies a Ricatti equation φ x + 2 W φ + φ = 0 . Through the standard variable transformation φ = q ′ /q , the function q is foundto satisfy a linear homogeneous equation q ′′ + 2 W q ′ = 0 . Utilizing the W expression in (27), we obtain the general q solution as q = ˆ c (cid:20) c + Z x [ ψ (1) ( ξ )] dξ (cid:21) , where c and ˆ c are arbitrary complex constants. In view of the variable trans-formation φ = q ′ /q , we see the constant ˆ c does not contribute to the φ solution.Putting all the above results together, we find the general function f W ( x ) as f W ( x ) = − ddx ln( e ψ (1) ) , (31)12here e ψ (1) ( x ) = ψ (1) ( x ) c + R x [ ψ (1) ( ξ )] dξ . For the new V factorization (30), its partner potential, defined through − d dx − e V ( x ) + µ (1) = (cid:20) − ddx + f W ( x ) (cid:21) (cid:20) ddx + f W ( x ) (cid:21) , is e V = V + 2 f W x . Utilizing the V and f W formulae (29) and (31), this e V potential is then foundto be e V ( x ) = V ( x ) + 2 d dx ln (cid:20) c + Z x [ ψ (1) ( ξ )] dξ (cid:21) . (32)For generic values of the complex constant c , this e V potential is complex andnon- PT -symmetric. In addition, its spectrum is identical to that of V . Indeed,even though µ (1) may not lie in the spectrum of V , it is in the spectrum of e V with eigenfunction e ψ (1) . Hence, if V has an all-real spectrum, so does e V .Notice that this e V potential, referred to as the superpotential below, is actuallya family of potentials due to the free complex constant c .Now we give two explicit examples of non- PT -symmetric superpotentials(32) with all-real spectra. The first one is constructed from the parabolic po-tential V ( x ) = − x and its first eigenmode of µ = − ψ = e − x / . Thenthe superpotential (32) reads V ( x ) = − x + 2 d dx ln (cid:20) c + Z x e − ξ dξ (cid:21) . (33)This potential with c = 1+i is shown in Fig. 4(a). The spectrum of this potential(for any complex c value) is {− , − , − , . . . } , i.e., is all-real.In the second example, the superpotential (32) is built from the PT -symmetricperiodic potential V ( x ) = V e x and its Bloch mode ψ (1) = I ( V e i x ) witheigenvalue µ = −
1. Here V is a real constant, and I is the modified Besselfunction. The resulting periodic superpotential (32) reads V ( x ) = V e x + 2 d dx ln (cid:20) c + Z x I ( V e i ξ ) dξ (cid:21) . (34)For V = 1 and c = 0 . − c values) is the same as that ofthe original potential V ( x ) = V e x , i.e., µ = − ( k + 2 m ) , where the wavenum-ber k is in the first Brillouin zone, k ∈ [ − , m is any non-negativeinteger.If V ( x ) is a localized real potential, then SUSY allows to construct localizedcomplex superpotentials (32) with all-real spectra [27, 28].13 x s up e r p o t e n t i a l V ( x ) (a)
0 −0.500.5 x (b) − π π Figure 4: (a) Superpotential (33) with c = 1 + i; (b) Periodic superpotential(34) with V = 1 and c = 0 . − V ), and the dashedred curve is Im( V ). Adapted from [28].A related technique to construct complex potentials with all-real spectra wasproposed by Cannata, et al. [25]. This technique is based on the formulae (27)and (29). But instead of choosing µ (1) as an eigenvalue of the potential V ( x )and ψ (1) as the corresponding eigenfunction, one chooses µ (1) as an arbitraryreal number and the function ψ (1) as a complex linear combination of the twofundamental solutions to the Schr¨odinger equation (25) [here we do not require ψ (1) to be square-integrable]. For instance, if V ( x ) is a real potential and µ (1) is an arbitrary real number, then we can choose ψ (1) as a linear combination c f ( x ) + c f ( x ), where f , f are the two real fundamental solutions to Eq.(25), and c , c are complex constants. With such choices of µ (1) and ψ (1) , it iseasy to see that the potential V ( x ) and the complex potential V ( x ) [as given byEq. (29)] still share the same spectrum in general. The only possible exceptionis regarding µ (1) . If 1 /ψ (1) is square-integrable, then since (cid:20) − ddx + W ( x ) (cid:21) ψ (1) = 0 ,µ (1) is in the discrete spectrum of V ; but it may not be in the discrete spectrumof V . Using this construction, non- PT -symmetric complex potentials with all-real spectra can also be obtained. For examples, see [25].One more variation of SUSY is based on the following observation. It canbe seen from Eqs. (26) and (28) that, for any complex functions W ( x ) and acomplex constant c , the two potentials − V ( x ) = W ( x ) − W ′ ( x ) + c, − V ( x ) = W ( x ) + W ′ ( x ) + c, form partner potentials which share the same spectrum (with the possible ex-ception of a single bound state). Thus, if we choose W ( x ) so that V ( x ) isreal, then the resulting complex potential V ( x ) will have an all-real spectrum.14hese complex potentials V ( x ) turn out to be type-II potentials (12) describedin Sec. 2.2 but with c ≤
0. An equivalent derivation of this result was givenby Andrianov et al. [16] and reviewed in the end of Sec. 2.3.
P T -symmetric potentialswith all-real spectra using soliton theory
Another technique to construct complex potentials with all-real spectra is touse the soliton theory. This technique was proposed by Wadati [15] for theconstruction of PT -symmetric potentials with all-real spectra, but it apparentlycan be extended to construct non- PT -symmetric potentials with all-real spectra,as we will demonstrate below.Let us consider the modified Korteweg-de Vries (mKdV) equation u t + 6 u u x + u xxx = 0 (35)for the real function u ( x, t ), where x is the spatial coordinate, and t is time.We will consider localized solutions, lim | x |→∞ u ( x, t ) = 0. Equation (35) is thecompatibility condition between the Zakharov-Shabat (ZS) spectral problem[17] v x + i ζv = u ( x, t ) v , v x − i ζv = − u ( x, t ) v , (36)and the linear system v t = 2i ζ ( u − ζ ) v + (2i ζu x − u − u xx + 4 ζ u ) v ,v t = (2i ζu x + 2 u + u xx − ζ u ) v − ζ ( u − ζ ) v . Here, ζ is a spectral parameter.The ZS spectral problem (36) can be transformed into Schr¨odinger eigenvalueproblems through the transformation (14). Under this transformation, we get ψ xx + V ( x, t ) ψ = µψ, φ xx + V ∗ ( x, t ) φ = µφ, (37)where V ( x, t ) = u ( x, t ) + iu x ( x, t ) , (38)and µ = − ζ . Here, time t plays the role of a parameter. If u ( x, t ) is an evenfunction of x , then the potential V ( x, t ) is PT symmetric; for general u ( x, t )solutions, this potential is complex and non- PT -symmetric.Discrete eigenvalues of the ZS problem (36) appear as quadruples ( ζ, ζ ∗ , − ζ, − ζ ∗ )if ζ is complex and as complex-conjugate pairs ( ζ, ζ ∗ ) if ζ is purely imaginary.The continuous spectrum of the ZS problem is the real- ζ axis. In view of theabove connection between the ZS and Schr¨odinger eigenvalue problems, we seethat from any solution u ( x, t ) of the mKdV equation (35) that possesses purelyimaginary discrete ZS eigenvalues, one can obtain a complex potential V ( x, t ),defined by (38), with strictly real spectrum. Further, we notice that while u ( x, t )depends on the parameter t , its ZS spectrum does not since u ( x, t ) satisfies the15KdV equation. This means that t can be considered as a “deformation” pa-rameter, and u ( x, t ) generates a family of deformable potentials V ( x, t ) withthe same real spectrum. Since the solution u ( x, t ) is asymmetric in general, theresulting complex potential V ( x, t ) is then non- PT -symmetric.Analytical solutions u ( x, t ) with purely imaginary discrete ZS eigenvalues canbe derived by the soliton theory. Indeed, through the inverse scattering method, N -solitons of the mKdV equation with purely imaginary discrete eigenvalues {± ζ n , ≤ n ≤ N } were found as [29] u ( x, t ) = − ∂∂x arctanImdet( I + A )Redet( I + A ) , (39)where I is the N × N identity matrix, Re and Im represent the real and imaginaryparts, A is the N × N matrix whose elements are A nm ( x, t ) = − c n ζ n + ζ m e i( ζ n + ζ m ) x +8i ζ n t ,ζ n = i η n , η n >
0, and c n are real constants. The corresponding complexpotential V ( x, t ) from Eq. (38) then would have all-real spectrum, with discreteeigenvalues as − ζ n = η n (1 ≤ n ≤ N ) and the continuous spectrum as ( −∞ , PT -symmetric potentials obtained from thetwo-soliton solution of the mKdV equation. These two solitons are found fromthe above general formula by taking N = 2 and can be written as [29] u ( x, t ) = 4 η + η η − η G ( x, t ) F ( x, t ) , (40)where G ( x, t ) = ǫ η cosh (cid:20) η x + δ ( t ) + 12 γ (cid:21) + ǫ η cosh (cid:20) η x + δ ( t ) − γ (cid:21) ,F ( x, t ) = cosh[2( η + η ) x + δ ( t ) + δ ( t )] + η η ǫ ǫ ( η − η ) + (cid:16) η + η η − η (cid:17) cosh[2( η − η ) x + δ ( t ) − δ ( t ) + γ ] ,ǫ = ± , ǫ = ± , η > , η > ,δ ( t ) = δ − η t, δ ( t ) = δ − η t, γ = ln( η /η ) , and δ , δ are real constants. To illustrate, we take η = 1 , η = 2 , δ = δ = 0 , ǫ = ǫ = 1 . (41)The soliton u ( x, t ), the complex potential V ( x, t ) and its spectrum at times t = 0 and 0.1 are displayed in the upper and lower rows of Fig. 5 respectively.Both complex potentials are non- PT -symmetric and differ from each other sig-nificantly, but they have identical real spectra.16 x u −5 0 5−4048 x V −5 0 5−101 Re( µ ) I m ( µ ) −5 0 504 x u −5 0 5−80816 x V −5 0 5−101 Re( µ ) I m ( µ ) Figure 5: Non- PT -symmetric potentials with real spectra from the soliton the-ory. Left column: the two-soliton solution (40) with parameters (41); middlecolumn: the corresponding complex potential V ( x, t ) from Eq. (38); right col-umn: spectrum of this V ( x, t ) potential. Upper row: t = 0; lower row: t = 0 . P T -symmetric potentials in multi-dimensions
In this section, we consider the generalization of PT symmetry to higher spatialdimensions. Let us consider a (2+1)-dimensional generalization of the paraxiallinear beam propagation equation (1),iΨ z + Ψ xx + Ψ yy + V ( x, y )Ψ = 0 , (42)where z is the propagation direction, and ( x, y ) is the transverse plane. Lookingfor eigenmodes of the form Ψ( x, y, z ) = e i µz ψ ( x, y ) we arrive at the eigenvalueproblem ( ∂ xx + ∂ yy + V ) ψ = µψ, (43)where µ is the eigenvalue and ψ the eigenfunction.The usual PT symmetry of the complex potential V ( x, y ) is defined as V ∗ ( x, y ) = V ( − x, − y ) , (44)i.e., the potential is invariant under complex conjugation and simultaneous re-flections in both x and y directions. For these potentials, the spectrum canbe all-real, with a possibility of phase transition, just like one-dimensional PT -symmetric potentials. 17owever, this usual concept of PT symmetry can be generalized. Indeed, ifthe potential is invariant under complex conjugation and reflection in a single spatial direction, i.e., V ∗ ( x, y ) = V ( − x, y ) , or V ∗ ( x, y ) = V ( x, − y ) , (45)its spectrum can still be all-real with a possibility of phase transition. Thesepotentials were introduced in [30] and termed partially- PT -symmetric poten-tials.The fundamental reason these partially- PT -symmetric potentials can alsofeature all-real spectra is that, their eigenvalues also come in complex conjugatepairs ( µ, µ ∗ ). This eigenvalue symmetry is a common feature of PT -symmetricpotentials, partially- PT -symmetric potentials, and complex potentials derivedin sections 2 and 3, which results in the possibility of all-real spectra for allthese potentials.The complex-conjugate-pair eigenvalue symmetry for these partially- PT -symmetric potentials is easy to prove. Indeed, if V ∗ ( x, y ) = V ( − x, y ) or V ∗ ( x, y ) = V ( x, − y ), then by taking the complex conjugate of Eq. (43) andswitching x → − x or y → − y , we see that µ ∗ would also be an eigenvalue witheigenfunction ψ ∗ ( − x, y ) or ψ ∗ ( x, − y ).As an example, we take the partially- PT -symmetric complex potential V ( x, y )to be localized at four spots: V ( x, y ) = 3 (cid:16) e − ( x − x ) − ( y − y ) + e − ( x + x ) − ( y − y ) (cid:17) +2 (cid:16) e − ( x − x ) − ( y + y ) + e − ( x + x ) − ( y + y ) (cid:17) +i β h (cid:16) e − ( x − x ) − ( y − y ) − e − ( x + x ) − ( y − y ) (cid:17) + (cid:16) e − ( x − x ) − ( y + y ) − e − ( x + x ) − ( y + y ) (cid:17)i , (46)where x , y control the separation distances between these four spots, and β is areal constant. For definiteness, we set x = y = 1 .
5. This potential is not PT -symmetric, but is partially- PT -symmetric with symmetry V ∗ ( x, y ) = V ( − x, y ).For β = 0 .
1, this potential is displayed in Fig. 6 (top row). It is seen thatRe( V ) is symmetric in x , Im( V ) anti-symmetric in x , and both Re( V ), Im( V )are asymmetric in y . The spectrum of this potential is plotted in Fig. 6(c). It isseen that this spectrum contains three discrete eigenvalues and the continuousspectrum, which are all-real.For potential (46) with varying β , we have found that its spectrum is all-realas long as | β | is below a threshold value of 0 . β = 0 . y (a) −6 0 6−606 xy (b) −6 0 6−606 −0.1 0 0.1 −2 0 2−101 Re( λ ) I m ( λ ) (c) β =0.1 −2 0 2−101 Re( λ ) I m ( λ ) (d) β =0.3 Figure 6: (a,b) Real and imaginary parts of the partially- PT -symmetric po-tential (46) for β = 0 .
1; (c, d) spectrum of this potential for β = 0 . Summary and discussion
In this article, we have reviewed various approaches to generalize PT symmetry.We have shown that large classes of non- PT -symmetric complex potentials canalso feature all-real spectra. These potentials are constructed by a variety oftechniques, such as the symmetry method, the supersymmetry method, the soli-ton theory and partial PT symmetry. Of these non- PT -symmetric potentials,the ones derived from the symmetry condition (4) in sections 2 and 3 allowfor arbitrary gain-loss profiles. In addition, as free parameters and functions inthose potentials vary, the spectrum could change, and phase transition (eitherthrough exceptional points or without) can occur. In non- PT -symmetric po-tentials derived from supersymmetry and the soliton theory, on the other hand,the gain-loss profile is not totally free; and as free parameters in those potentialsvary, the spectrum stays exactly the same.The focus of this article was the spectrum of non- PT -symmetric complex po-tentials, which is inherently a linear theory. When the spectrum of the complexpotential is all-real, then wave propagation in the linear evolution equations (1)and (42) would show features which resemble those in real potentials (withoutgain and loss). When nonlinearity arises in these complex potentials, wherenonlinear terms appear in the evolution equations (1) and (42), the interplaybetween nonlinearity and these complex potentials is an interesting question.In PT -symmetric potentials and other PT -symmetric systems, this interplaybetween nonlinearity and PT symmetry has been reviewed in [12, 31]. In non- PT -symmetric potentials, it was shown that the evolution equation (1) withKerr nonlinearity could admit continuous families of solitons in type-I poten-tials (9), but not in other types of complex potentials [14, 28, 32, 33]. Howother types of nonlinearities interact with these complex potentials is a worthyquestion for study in the future. Acknowledgment
This material is based upon work supported by the Air Force Office of Scien-tific Research under award number FA9550-18-1-0098, and the National ScienceFoundation under award number DMS-1616122.
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