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Scattering Theory and \(\mathbb {P}\mathbb {T}\)-Symmetry

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Parity-time Symmetry and Its Applications

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 280))

Abstract

We outline a global approach to scattering theory in one dimension that allows for the description of a large class of scattering systems and their \(\mathbb {P}\)-, \(\mathbb {T}\)-, and \(\mathbb {P}\mathbb {T}\)-symmetries. In particular, we review various relevant concepts such as Jost solutions, transfer and scattering matrices, reciprocity principle, unidirectional reflection and invisibility, and spectral singularities. We discuss in some detail the mathematical conditions that imply or forbid reciprocal transmission, reciprocal reflection, and the presence of spectral singularities and their time-reversal. We also derive generalized unitarity relations for time-reversal-invariant and \(\mathbb {P}\mathbb {T}\)-symmetric scattering systems, and explore the consequences of breaking them. The results reported here apply to the scattering systems defined by a real or complex local potential as well as those determined by energy-dependent potentials, nonlocal potentials, and general point interactions.

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Notes

  1. 1.

    These are occasionally labelled by T l∕r(k) and R l∕r(k), [44, 59]. Here we refrain from using this notation, because some references use these symbols for the reflection and transmission amplitudes and not their modulus squared [25].

  2. 2.

    By a scatterer we mean the interaction causing the propagation of a wave differ from that of a plane wave.

  3. 3.

    In Sect. 4, we prove that this conditions holds for the scattering systems described by the Schrödinger equation (2).

  4. 4.

    A genuine multidimensional generalization of the transfer matrix has been recently proposed in [31].

  5. 5.

    This can be easily checked by differentiating W(x) and using (2) to show that W ′(x) = 0.

  6. 6.

    Equation (58) implies that \(k_m=\sqrt {(\pi m/L)^2+\mathfrak {z}}\). This in turn means that for \(\mathfrak {z}>0\), m can be any positive integer, and for \(\mathfrak {z}<0\), \(m>L\sqrt {-\mathfrak {z}}/\pi \).

  7. 7.

    Reflectionless potentials also arise as soliton solutions of nonlinear differential equations [24].

  8. 8.

    This means that v(x) is unidirectionally left-invisible for k = K n∕2 = πn∕L provided that we can neglect terms of order \((\mathfrak {z}_n/k^2)^2\) in the calculation of the reflection and transmission amplitudes.

  9. 9.

    For a review of basic properties of these potentials, see [15].

  10. 10.

    This explains why Theorems 6 and 8 do not conflict.

  11. 11.

    The notion of a spectral singularity was originally introduced in [60] for Schrödinger operators in the half-line. It was subsequently generalized to the case of full-line in [21]. The term “spectral singularity” was originally used to refer to this notion in [69]. For a readable account of basic mathematical facts about spectral singularities and further references, see [14].

  12. 12.

    Spectral singularities must be distinguished with the solutions of the time-independent Schrödinger equation that correspond to a bound state in the continuum [17, 72] for the following reasons: (1) They define scattering states that do not decay at spatial infinities. (2) They may exist for exponentially decaying and short-range potentials. (3) As we explain in Sect. 8, real potentials cannot have spectral singularities. None of these holds for bound states in the continuum.

  13. 13.

    The importance of purely outgoing waves in the laser theory predates the discovery of their connection to the mathematics of spectral singularities. See for example [73].

  14. 14.

    For a proof of this statement see [39, Appendix]. A more detailed discussion is provided in [68].

  15. 15.

    This is obviously not always possible. A sufficient condition for the existence of such a modified inner product is that the operator L satisfies the pseudo-Hermiticity relation L † = η Lη −1 for a positive-definite bounded linear operator η with a bounded inverse. For further discussion of these and related issues see [39, 45] and references therein.

  16. 16.

    A simple examples is \(\mathbb {T}_\tau :=e^{i\tau }\mathbb {T}\) where \(\tau \in \mathbb {R}\).

  17. 17.

    The scattering problem for this Hamiltonian operator is equivalent to that of the energy-dependent scattering potential \(v(x,k):=2m v(x)+ k^2/(1+e^{\mu x^2})\). This is because we can write Hψ(x) = Eψ(x) in the form − ψ ′′(x) + v(x, k)ψ(x) = k 2 ψ(x) where \(k:=\sqrt {E}\).

  18. 18.

    An extension of this theorem to more general scattering systems is given in [52].

  19. 19.

    Equations (164) was originally conjectures in [1] for \(\mathbb {P}\mathbb {T}\)-symmetric scattering potentials based on evidence provided by the study of a complexified Scarf II potential. It was subsequently proven in [48] for general \(\mathbb {P}\mathbb {T}\)-symmetric scattering potentials which respect transmission reciprocity.

  20. 20.

    For a 2 × 2 matrix A, the condition of being σ 1-pseudo-unitary is equivalent to the requirement that \(e^{i\pi \boldsymbol {\sigma }_2/4}\mathbf {A} e^{-i\pi \boldsymbol {\sigma }_2/4}\) belong to the pseudo-unitary group U(1, 1), where σ 2 is the second Pauli matrix.

  21. 21.

    We use the term “eigenvalue” to mean an element of the point spectrum of H which has a square-integrable eigenfunction.

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Acknowledgements

The author is indebted to Keremcan Doğan, Sasan HajiZadeh, and Neslihan Oflaz for their help in locating a large number of typos in the first draft of the manuscript. This work has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) in the framework of the project no: 114F357 and by Turkish Academy of Sciences (TÜBA).

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  1. Departments of Mathematics and Physics, Koç University, Istanbul, Turkey

    Ali Mostafazadeh

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  1. College of Optics and Photonics, University of Central Florida, Orlando, FL, USA

    Demetrios Christodoulides

  2. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

    Jianke Yang

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Mostafazadeh, A. (2018). Scattering Theory and \(\mathbb {P}\mathbb {T}\)-Symmetry. In: Christodoulides, D., Yang, J. (eds) Parity-time Symmetry and Its Applications. Springer Tracts in Modern Physics, vol 280. Springer, Singapore. https://doi.org/10.1007/978-981-13-1247-2_4

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