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Exceptional points and pseudo-Hermiticity in real potential scattering

by Farhang Loran and Ali Mostafazadeh

Submission summary

As Contributors: Ali Mostafazadeh
Preprint link: scipost_202109_00035v1
Date submitted: 2021-09-29 20:09
Submitted by: Mostafazadeh, Ali
Submitted to: SciPost Physics
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator $\widehat\boldsymbol{H}$ which develops an exceptional point for a discrete set of incident wavenumbers. We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem. We apply our general results to explore the scattering of waves by a waveguide of finite length in two dimensions, where the source of the incident wave and the detectors measuring the scattered wave are positioned at spatial infinities while the interior of the waveguide, which is filled with an inactive material, forms a finite rectangular region of the space. The study of this model allows us to elucidate the physical meaning and implications of the presence of the real and complex eigenvalues of $\widehat\boldsymbol{H}$ and its exceptional points. Our results reveal the relevance of the concepts of pseudo-Hermitian operator and exceptional point in the standard quantum mechanics of closed systems where the potentials are required to be real.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 1 on 2021-11-23 (Invited Report)


1. The manuscript is written in a clear, detailed, and self-contained manner.

2. The general results are well supported by the detailed investigation of a specific example.

3. The discovery that exceptional points are relevant even to the scattering problem of real potentials is interesting and may lead to further applications.


1. The readability of the manuscript may be low for general readers (the manuscript is written in a self-contained manner, though).


The authors develop a scattering theory in two dimensions for a class of potentials described by Eq. (1). They demonstrate that the transfer matrix for this scattering problem effectively reduces to a time-evolution operator for a non-Hermitian Hamiltonian with pseudo-Hermiticity even though the original scattering problem is concerned with a Hermitian Hamiltonian. As an example, they apply their general results to a specific waveguide of finite length. For this example, they show that the intensity of the transmitted wave is invariant under the continuous change of the length of the waveguide as a consequence of an exceptional point in the effective non-Hermitian Hamiltonian.

In my humble opinion, this manuscript discovers a new important role of exceptional points in the scattering theory, which opens a new pathway in non-Hermitian physics. Thus, I believe that this manuscript meets the acceptance criteria of SciPost Physics and I would like to recommend publication of this manuscript in SciPost Physics.

Requested changes

I have a couple of relatively minor comments, explained below.

1. As pointed out in the first observation in Sec. 3 (page 8), the spectrum of the effective non-Hermitian Hamiltonian consists of pairs of eigenvalues with opposite signs. This property should not originate from pseudo-Hermiticity since pseudo-Hermiticity only leads to real eigenvalues or complex-conjugate pairs of eigenvalues. Then, what symmetry ensures the opposite-sign pairs of eigenvalues? It seems to me that this structure originates from another internal symmetry such as particle-hole symmetry and sublattice symmetry [please see, for example, K. Kawabata et al., Phys. Rev. X 9, 041015 (2019) for details on internal symmetry of non-Hermitian Hamiltonians].

2. As described below Eq. (139) in Sec. 5 (page 20), a physical consequence of exceptional points in the scattering theory is the invariance of the intensity of the transmitted wave under the continuous change of the length of the waveguide. Is this property unique to exceptional points in the effective non-Hermitian Hamiltonian? In other words, can we realize the same phenomenon even if the effective Hamiltonian does not support exceptional points? Or, is it impossible to realize this phenomenon if the effective Hamiltonian does not support exceptional points?

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: reasonable
  • grammar: reasonable

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Anonymous on 2021-11-27  [id 1981]

answer to question

In the following we respond to the questions posed by the referee in her/his report.

1.~The referee asks about the origin of the structure of the spectrum of the effective non-Hermitian Hamiltonian~(33), namely the fact that its eigenvalues come in pairs of oppose sign. It is not difficult to show that the Pauli marix $\boldsymbol{\sigma}_1$ anti-commutes with the Hamiltonian, i.e., $\{\boldsymbol{\sigma}_1,\widehat{\mathbf{H}}\}={{\boldsymbol{0}}}$. Alternatively, according to (44), $\boldsymbol{\sigma}_1|\Psi_{n,\pm}{\rangle}=|\Psi_{n,\mp}{\rangle}$, which means that $\boldsymbol{\sigma}_1$ swaps the eigenvectors with eigenvalues differing by a sign. Therefore, if we use the terminology of K.~Kawabata et al, Phys. Rev. X {\bf 9}, 041015 (2019), we would say that this property of the spectrum of $\widehat{\mathbf{H}}$ follows from the chiral symmetry characterized by $\{\boldsymbol{\sigma}_1,\widehat{\mathbf{H}}\}={{\boldsymbol{0}}}$. This terminology does not however agree with the standard concept of symmetry (due to Wigner) which is described in terms of the commutation of ${\mathbf{H}}$ with either a linear or an antilinear operator. In view of Theorem~2 of Ref.~[38], the structure of the spectrum of $\widehat{\mathbf{H}}$ shows that both $\widehat{\mathbf{H}}$ and $i\widehat{\mathbf{H}}$ are pseudo-Hermitian and that each of them must commute with an antilinear involution. This implies the existence of antilinear involutions $\widehat{\mathfrak{S}}$ and $\widehat\chi$ satisfying
We can use the constructions given in Ref.~[38] to obtain spectral series expansions for $\widehat{\mathfrak{S}}$ and $\widehat\chi$. We do not report these in our paper, because we have not been able to find a useful physical application for them. We plan to add a remark on these observations in a revised version of our manuscript.

2.~The referee asks whether the invariance of the intensity of transmitted wave under continuous changes of the length of the waveguide is an exclusive consequence of the presence of an exceptional point. The answer to this question is in the affirmative. This can be seen from Eq.(137). For cases where there is no exceptional point, $w_{n}\neq 0$ for all $n$, and all the terms contributing to $\Gamma_+(p,p_0)$ become $a$-dependent.