Physics Letters A 377 (2013) 957–959

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Physics Letters A

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Reciprocity and unitarity in scattering from a non-Hermitian complexPT-symmetric potential

Zafar Ahmed

Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 December 2012Received in revised form 11 February 2013Accepted 15 February 2013Available online 20 February 2013Communicated by P.R. Holland

In quantum scattering, Hermiticity is necessary for both reciprocity and unitarity. Reciprocity means thatboth reflectivity (R) and transmitivity (T ) are insensitive to the direction of incidence of a wave (particle)at a scatterer from left/right. Unitarity means that R + T = 1. In scattering from non-Hermitian PT-symmetric structures the (left/right) handedness (non-reciprocity) of reflectivity is known to be essentialand unitarity remains elusive so far. Here we present a surprising occurrence of both reciprocity andunitarity in some parametric regimes of scattering from a complex PT-symmetric potential. In specialcases, we show that this potential can even become invisible (R = 0, T = 1) remarkably this time fromboth left and right sides. We also find that this potential in a parametric regime enjoys a pseudo-unitarityof the type: T + √

Rleft Rright = 1.© 2013 Elsevier B.V. All rights reserved.

In non-relativistic quantum mechanics Hermiticity is the neces-sary condition for a Hamiltonian to have: real discrete spectrum,and both unitarity and reciprocity in scattering. Reciprocity meansthat both reflectivity (R) and transmitivity (T ) are insensitive tothe direction of incidence of a wave (particle) at a scatterer fromleft/right. Unitarity in scattering means that R + T = 1. In variousbranches of physics the complex optical potentials have been inuse since a long time to account for the absorption of the inci-dent flux in to unknown channels. Consequently, non-Hermiticityis synonymous to absorption or emission of flux. In this kind ofscattering the unitarity is broken as the probability of reflection(R) and transmission (T ) do not add to 1 and one instead hasR + T + A = 1 where A is the probability of absorption.

Bender and Boettcher [1,2] conjectured that the eigenspectrumof a non-Hermitian complex potential in a parametric regime wasdiscrete and real. This potential was PT-symmetric [invariant un-der Parity (x → −x) and Time-reversal (i → −i)]. Also this poten-tial was not amenable to exact analytic solutions so it requiredspecial methods to prove the reality of its spectrum [3]. Theirconjecture has initiated a debate: ‘Must a Hamiltonian be Hermi-tian?’ [2] and it has inspired a large body of investigations leadingto the extension of the quantum mechanics in complex domain(see e.g., [1–21,23–31]). About thirteen years later, the presentwork addresses the same question but this time for reciprocityand unitarity in scattering. Surprisingly, this time again the answeris no.

E-mail address: zahmed@barc.gov.in.

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2013.02.031

For the scattering from a complex non-Hermitian potential ithas been possible to prove [4] that if non-Hermitian complex po-tential is spatially asymmetric the reflectivity (R) shows handed-ness Rleft �= Rright whereas transmitivity (T ) remains invariant tothe direction of the incidence of the particle from left or right. Thecomplex PT-symmetric potentials being spatially anti-symmetricare automatically entitled to this handedness [5–18]. This con-trasting feature of scattering from that of reciprocity in Hermitiancase perhaps may have discouraged one to look for unitarity inthe scattering from complex PT-symmetric potentials. Nevertheless,various works [4–18] normally display non-unitarity in scatteringfrom complex PT-symmetric potentials.

There has been a very impressive progress in the investiga-tions of the scattering from a complex PT-symmetric potential.In some PT-symmetric structures the absence unitarity has beenmarked with new pseudo-unitarity conditions such as T − 1 =±√

Rleft Rright ,Rleft+Rright

2 − T = 1 (see Eqs. (9) and (17), respec-tively in Ref. [9]). The concepts like spectral singularity [11–14]and invisibility [15,18] have been well developed both theoreti-cally and experimentally. For the spectral singularity one looks forpositive energies where there are very large (infinite) [11] peaksin both R(k) and T (k). The instance where Rleft(k) or Rright(k) = 0and T (k) = 1 is called unidirectional invisibility [15–18] in bothcomplex-non-Hermitian and complex PT-symmetric potentials. No-tice that this invisibility is direction dependent either from left orfrom right. This is the consequence of the handedness of reflectiv-ity in these potentials.

The ghost of non-Hermiticity has already been busted in PT-symmetric domain, new features such as spectral singularity

958 Z. Ahmed / Physics Letters A 377 (2013) 957–959

[11–14] and invisibility [15–17] of such potentials are beinginvestigated. Novel optical devices and materials have been en-gineered to realize wave propagation through a complex PT-symmetric medium [9,15–17,19–21]. In this scenario of scatteringfrom a complex PT-symmetric potential, we present surprisingparametric regimes in the well-known complex PT-symmetricScarf II potential wherein we observe reciprocity, unitarity, invis-ibility (from both left and right). We also find that this poten-tial satisfies one of the newly proposed pseudo-unitarity condi-tions [9].

The scattering from Scarf II potential is well studied whereinthe reflection and transmission amplitudes have been well workedout [5,22]. The non-Hermitian complex PT-symmetric version ofthe Scarf II potential has been very useful in the investigationsof complex PT-symmetric potentials in various ways [5,7,13,14,23–30]. We would like to write the complex Scarf II potential as

V (x) = −(B2 + A2 + A

)sech2 x + iB(2A + 1) tanh x sech x, (1)

which is known to have real discrete spectrum [5,22–28]. This po-tential in another parametric form displays phase-transition [24] ofreal discrete eigenvalues to complex conjugate pairs about a criti-cal value of a parameter when the PT-symmetry breaks down [1,2].Let 2μ = 1 = h̄2 and k = √

E , where E is the energy. Following [5,22], we can write the transmission amplitude for (1) as [30]

t A,B(k) = Γ [−A − ik]Γ [1 + A − ik]Γ [ 12 + B − ik]Γ [ 1

2 − B − ik]Γ [−ik]Γ [1 − ik]Γ 2[ 1

2 − ik] ,

(2)

rA,B(k) = t A,B(k)i

[cosπ A sinπ B

coshπk+ sinπ A cosπ B

sinhπk

]. (3)

The transmitivity T (k) = |t(k)|2 and the reflectivity R(k) = |r(k)|2.We have re-derived (2)–(3) to find [14] that for (1)

tleft(k) = t A,B(k), rleft(k) = rA,B(k) and

tright(k) = t A,−B(k), rright(k) = rA,−B(k). (4)

However, this point can also be verified easily by noticing thatthe potential (1) satisfies: V (x, B) = V (−x,−B). Making multipleuse of the property of Gamma functions namely Γ (z)Γ (1 − z) =π cosecπ z we express the transmitivity, T (k) as

T (k) = sinh2 πk cosh2 πk

(sinh2 πk + sin2 π A)(sinh2 πk + cos2 π B), A, B ∈ R.

(5)

It follows that T (k) will be both normal (< 1) and anomalous(> 1). For the cases A = B or when A = n+1/2, or B = n, n ∈ I , thetransmitivity is normal. For the cases A = n and A �= B the trans-mitivity is anomalous at small energies. Moreover, when the trans-mitivity (5) is normal, it can be readily checked that the presentresults on T (E) and Rleft(E) and Rright(E) (2)–(5) satisfy a pseudo-unitarity of the type:

T (E) +√

Rleft(E)Rright(E) = 1. (6)

See Ref. [9] for this (6) and other proposals of pseudo-unitarity. Or-dinarily, for real values of the parameters A, B Eqs. (2)–(5) yield tothe rule [4,8,9,11,15,16] of left/right handedness (non-reciprocity)of R(k) and sometimes the non-unitarity manifests as a pseudo-unitarity condition (6). For other (in)variances see [14]. WhenA = −(n+1)− iα and B = iα−(n+1/2) with n ∈ I+ +{0}, α > 0 inEqs. (2)–(3) a recent phenomenon of spectral singularity [11] is ob-served wherein at E = α2 [14] both R and T become infinite. Forother special values of the parameters A and B , from Eqs. (2)–(5)the following extra-ordinary features arise:

{1} Reciprocity and unitarity. Case 1: When A = n+1/2, n ∈ I , andB is real, from (2) and (3) we get

R(k) = cos2 π B

sinh2 πk + cos2 π B, T (k) = sinh2 πk

sinh2 πk + cos2 π B.

(7)

Case 2: When B ∈ I and A is real, we get

R(k) = sin2 π A

sinh2 πk + sin2 π A, T (k) = sinh2 πk

sinh2 πk + sin2 π A.

(8)

In both the cases from Eq. (4) the acclaimed reciprocity of reflec-tivity follows: Rleft(k) = Rright(k). The reflectivity is also symmetricunder time-reversal: R(−k) = R(k). The claimed unitarity can bechecked readily using Eqs. (7)–(8). �{2} Invisibility with reciprocity. In the above two cases of uni-tarity when A = (n + 1/2), B = (m + 1/2) or A = n, B = m(n,m ∈ I) check that two cases of invisibility occur whereinR(k) = 0, T (k) = 1 at any energy. This invisibility of the complexPT-symmetric potential (1) is not unidirectional [15–18], this timeit is from both sides left and right. �

Earlier such an invisibility has been termed as bi-directional in-visibility and first found in V (x) = − 1

(x+ia)2 [31] using the methods

super symmetric quantum mechanics. More recently bi-directionalinvisibility has been found in the PT-symmetric Ginocchio’s poten-tial [32]. Interestingly, both these instances are for the potentialsof the type V (x) = V 0 f (x + ia) where a real Hermitian potentialhas been complexified by an imaginary shift of the x co-ordinate.In these types of potentials as argued in [5] Rleft, Rright are of thetype e±2ka R ′ where R ′ is reflectivity of the Hermitian potential(V 0 f (x)). Consequently, the pseudo-unitarity (6) is satisfied andthe left and right reflectivity zero(s) occur at the same discrete en-erg(ies).

The reflectivity of the non-Hermitian complex potentials whichare spatially symmetric shows [4] reciprocity along with non-unitarity R + T < (>) 1 when the imaginary part of the po-tential is negative (positive) definite for x ∈ (−∞,∞). For exam-ple for V (x) = (V 1 ∓ iV 2) sech2 x, V 1, V 2 ∈ R , V 2 > 0 (V 2 < 0),the reflectivity will show reciprocity along with non-unitarityR + T < (>) 1, respectively. Hence, we know that reciprocity doesnot imply unitarity. We speculate that unitarity may be sufficientfor reciprocity of reflectivity in scattering from a complex non-Hermitian potential.

Notwithstanding a rapid research in PT-symmetric structuresthese days and the familiarity of the complex PT-symmetric ScarfII potential, to the best of our knowledge, the above two para-doxical or exceptional features {1,2} are new and un-noticed, sofar. The observations and the proofs of the non-reciprocity of re-flectivity in scattering from complex PT-symmetric potentials areabound, however, one ought to look for reciprocity under somespecial parametric condition hereafter. The question whether thereare various parametric regimes in other PT-symmetric structuresyielding to reciprocity (of reflectivity), unitarity, spectral singular-ity, invisibility and the pseudo-unitarity of the type (6) is open forinvestigations. With regard to this, studying exactly solvable com-plex potentials becomes even more important. The present exposi-tion provides a paradigm shift in the thinking in two ways. Firstly,in scattering, Hermiticity and time-reversal symmetry of an inter-action are not necessary for unitarity and reciprocity (of reflectiv-ity), respectively. Secondly, the complex PT-symmetric structuresare very versatile having multiple parametric regimes displayingvarious properties.

Z. Ahmed / Physics Letters A 377 (2013) 957–959 959

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