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Exceptional points and pseudo-Hermiticity in real potential scattering

by Farhang Loran and Ali Mostafazadeh

This Submission thread is now published as SciPost Phys. 12, 109 (2022)

Submission summary

As Contributors: Ali Mostafazadeh
Preprint link: scipost_202109_00035v2
Date accepted: 2022-02-25
Date submitted: 2022-01-14 16:01
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.

Published as SciPost Phys. 12, 109 (2022)

Author comments upon resubmission

We had already provided details responses to the referees' remarks and questions. The editor has found these satisfactory, and as his/her initial decision on our paper, has asked for minor revisions. In preparing the revised manuscript, we have followed the editor's recommendations to incorporate the referees’ constructive suggestions along the lines we had outlined in our response to their comments.

We hope that with the changes we have made in our manuscript, it is now suitable for publication in SciPost.

List of changes

We have added comments and new material to Sec. 2 (the line below Eq. 14), Sec. 3 (2nd line below the numbered list on page 8 and Footnote 3), and Sec. 5 (Fig. 3, which demonstrates the behavior of the eigenvalues and exceptional points of the effective Hamiltonian H, and its discussion given below Eq. 112, and Line 4 on page 22), and a new reference (Ref. 40). We have marked all changes made in the manuscript in red.

Reports on this Submission

Anonymous Report 2 on 2022-2-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202109_00035v2, delivered 2022-02-16, doi: 10.21468/SciPost.Report.4437


1 - The paper presents a detailed, rigorous derivation of the authors' results
2 - It introduces a new approach to scattering theory, dealing with a specific class of 2D potentials
3 - It draws a connection between transport properties of systems without gain and loss and the non-Hermitian physics of exceptional points


1 - Due to its nature, the paper is highly technical, which makes it sometimes difficult to follow


The authors present a new approach to scattering theory, which applies to a range of two-dimensional problems for which the conventional scattering theory does not work. They use this framework to show a connection between transport properties in Hermitian systems and the exceptional points commonly associated with non-Hermitian systems, which would otherwise require gain and/or loss. The paper is detailed and self-contained, but it is highly technical, and it was difficult for me to follow the authors' derivation. However, I believe this is to be expected given the content of the paper and its results.

Given the above, my opinion is that this work does meet the acceptance criteria for publication in SciPost Physics ( Specifically, it opens a new pathway in an existing research direction, with clear potential for multipronged follow-up work.

I have no changes to request, but I do have one question, mostly for the sake of satisfying my own curiosity. I would imagine that it is possible to associate the transfer matrix with an effective time-evolution operator governed by a non-Hermitian Hamiltonian also when doing the conventional scattering theory of Lippmann and Schwinger. Do you expect exceptional points to occur also in that case, or would you rather expect them to be a feature of the two-dimensional potentials you consider in this work?

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Author:  Ali Mostafazadeh  on 2022-02-16  [id 2211]

(in reply to Report 2 on 2022-02-16)
answer to question

This referee has also found our work meeting the acceptance criteria for publication in SciPost Physics and requested no changes. (S)he has asked us the following question: “I would imagine that it is possible to associate the transfer matrix with an effective time-evolution operator governed by a non-Hermitian Hamiltonian also when doing the conventional scattering theory of Lippmann and Schwinger. Do you expect exceptional points to occur also in that case, or would you rather expect them to be a feature of the two-dimensional potentials you consider in this work?”

In response to this question, we wish to point out that It is absolutely correct that we can use our approach to deal with the (short-range) potentials to which the standard scattering theory of Lippmann and Schwinger applies, and that the corresponding effective Hamiltonian will be non-Hermitian. Although it is not easy to make general statements about the structure and spectral properties of this Hamiltonian, we expect it to possess exceptional points whenever it has a point spectrum. The study of these exceptional points and their physical implications is an interesting topic for future research. Our work provides the first step towards exploring this topic.

Anonymous Report 1 on 2022-1-19 (Invited Report)


I would appreciate the response and the corresponding revision of the manuscript. The authors have satisfactorily addressed the comments in my first report. Now, I would like to recommend publication of this manuscript in SciPost Physics.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Login to report


Uwe Guenther  on 2022-02-21  [id 2230]


The subsequent conceptual remark was part of an Invited Report finalized 4 hours after the Report deadline (end of the day of 16 Feb 2022) so that the SciPost reporting option was automatically closed and only comments were possible afterwards.


On page 2, end of paragraph 2 the authors state:

"The present investigation differs from the earlier works on the physical aspects of exceptional points in that it deals with exceptional points arising in the treatment of a scattering problem for a real potential. In the context of their optical or acoustic realizations, these are exceptional points whose presence does not require active or lossy materials."

This statement can be embedded into a slightly broader conceptual context, having a closer glance at the Feshbach projection technique [r1,r2] as applied to derive from Hermitian setups (Hamiltonians with real potentials) effective non-Hermitian setups. This basic technique was used, e.g., to study the hidden subtleties of nuclear shell models [r3] and, in this way, it has been one of the starting points of the earlier investigations of I. Rotter [r4] leading later to her investigations into setups with exceptional points (EPs). Similarly, Feshbach-projected S-matrices have been used as theoretical background for the PRL series of the Darmstadt microwave experimental group around A. Richter [r5,11,r6] as well as in earlier slightly related mathematical studies [r7].
(Implicitly and roughly/conceptually spoken, the Naimark dilation approach of [r8] can be considered as a kind of "inverted Feshbach projection" technique in extending/dilating a given non-Hermitian setup into an associated Hermitian setup living in a Hilbert space of higher dimension.)

With regard to the present work, one can observe a strong conceptual analogy to Feshbach projections in the sense that an initial 2D-scattering setup with purely real potential is mapped (roughly spoken "projected") to an effective 1D evolution problem governed by the transfer-matrix-related non-Hermitian Hamiltonian. Clearly, the transfer matrix approach described in the present work (and developed in a series of earlier publications of the second author) is a deep novel finding whose importance is in no way diminished by very rough conceptual analogies to Feshbach projection techniques as known since the late 1950s. Rather, it can be identified as another important realization of a deep conceptual scheme which might be dubbed, e.g., a "generalized Feshbach-like dynamical projection" associating to a given higher-dimensional Hermitian setup an effective lower-dimensional non-Hermitian setup.

[r1] H. Feshbach, Ann. Phys. 5, 357 (1958).

[r2] H. Feshbach, Ann. Phys. 19, 287 (1962).

[r3] C. Mahaux and H.A. Weidenmueller, "Shell-model approach to nuclear reactions",
(North-Holland, Amsterdam, 1969).

[r4] I. Rotter, Rep. Prog. Phys. 54, 635 (1991).

[r5] B. Dietz et al., Phys. Rev. Lett. 98, 074103 (2007).

[r6] S. Bittner et al., Phys. Rev. Lett. 108, 024101 (2012).

[r7] S. Albeverio et al., J. Math. Phys. (N.Y.) 37, 4888 (1996).

[r8] U. Guenther and B.F. Samsonov, Phys. Rev. Lett. 101, 230404 (2008).