Exceptional points and pseudo-Hermiticity in real potential scattering

This referee has expressed his assessment by filling the online assessment form of SciPost and also attaching a PDF of his/her report whose content is slightly different. His/Her recommendation in the assessment form reads: ``My recommendation is that the paper should be revised significantly before acceptance in a more technical journal.'' while in the PDF (s)he writes: ``My recommendation is that the paper should be revised significantly before acceptance because it is difficult to appreciate many of the presented results." This is in sharp contrast with the opinion of the first referee who had no problem appreciating the significance of these results and recommended the publication of our paper by scoring its clarity as ``high.''

The main criticism of the second referee is that we have included too many definitions and equations without explanations, and that we should have instead provided graphical demonstration of our results. We believe that for a basic contribution to scattering theory, such as the one we report in this paper, the number of equations or lack of nice-looking graphs should have no baring on the significance of the results. The same criticism can also be made about the ground-breaking 1950 paper of Lippmann and Schwinger [Phys. Rev. 79, 469-480 (1950)] which includes 156 equations and no figures or appendices. We would definitely consider removing any of equations or their derivations that the referee finds unnecessary had (s)he specified their equation numbers. In a future revision we will consider moving some of the mathematical derivations to appendices, if this will ease the readability of the manuscript and does not hurt its coherence.

We agree that we could have offered more detailed explanation of the meaning of some of the equations, but we consider the present form of the manuscript sufficiently detailed. This view is shared by the first referee who stresses in his/her report that our presentation is absolutely self-contained and clear. We wish to draw attention to the fact that, by the very nature of the subject, one needs to have sufficient background in quantum mechanics to follow and appreciate this work. This does not however imply that it is a technical development beyond the reach of a general theoretical physicist. Indeed, every serious student of theoretical physics with a working knowledge of quantum mechanics (say at the level of Sakurai's QM) should be able to follow the arguments and repeat the calculations we present in our paper. Obviously, this requires certain amount of time and effort. After all it took us several months of intensive work to solve the multitude of problems we encountered in pursuing this line of research.

As far as the significance of our findings are concerned, we wish to recall the following points which we have underlined in the abstract, introduction, and the conclusion of the paper.

(i) This paper is the first to reveal the relevance of the notions of ``pseudo-Hermitian operator'' and ``exceptional point'' in the scattering theory of real potentials.

(ii) It is the first to offer a complete analytic solution of the scattering problem for a waveguide with a finite length when the source of the incident wave and the detectors are not placed at the ends of the waveguide but at spatial infinities.

(iii) It is the first to explain a concrete physical effect related to the presence of an exceptional point in waveguide system with no regions of gain or loss.

We believe that these are significant developments on their own sake, not to mention that they follow from what this referee qualifies as ``a novel scattering-matrix formulation.'' Our results are not restricted to a waveguide system but to a large class of physically relevant scattering problems whose treatment is beyond the standard methods of scattering theory. We are surprised how the referee could ignore these points, and conclude that the significance of our work is ``low."

Both version of the second referee's report includes 10 numbered more specific remarks/questions. The following are our response to these remarks.

Remark 1. The referee asks: ``Are the exceptional points that you find related to the exceptional points associated with total internal reflection?''

Our response to Remark 1: We are not aware of any reference on exceptional points for a waveguide that does not include gain or loss regions. Therefore, we do not understand what the referee means by ``the exceptional points associated with total internal reflection.''

Remark 2. The referee recommends that we write a sequel to our previous work on scattering theory where we ``can state the main equations one should use in order to construct the relevant operators in order to apply the method – rather than derive many results that were already published and present more than 100 equations. recollect the steps.''

Our response to Remark 2: The referee is under the impression that in this paper we ``derive many results that were already published.'' A quick examination of the references to our earlier works (Refs. 32 and 33) can easily show that this claim is not true. The present paper is the first in which we apply the ideas developed in Ref. 33 to potentials that decay to zero at spatial infinity only along some directions and blow-up along others. Standard methods of scattering theory (which are based on Lippmann-Schwinger equation) are not capable of dealing with such potentials. We emphasize that we have not derived any results in this paper that is already published by us or others. We did precisely what the referee suggests, i.e., recollected the necessary steps of our earlier work (Ref. 33) that we needed in dealing with potentials of the form (1). This corresponds to the content of the text that appears between Eqs. (4) -- (31), excluding the discussion of the reflection and transmission amplitudes and the S-matrix [given in Eqs. (15), (16), and (19) -- (21)] which we present for the first time. Our review of the earlier results occupies less than 15\% of the volume of our manuscript. The rest of the paper reports absolutely original developments. This includes the very definition of the effective Hamiltonian (33) which possesses the exceptional points. Therefore, we do not understand how the referee has reached the conclusion that we ``derive many results that were already published and present more than 100 equations.''

Remark 3. The referee writes ``I think it would be good to mention the advantages of the current formulation over previous scattering-matrix formulations of two-dimensional problems that require integration over momenta.''

Our response to Remark 3: We believe the referee means ``transfer-matrix formulation'' rather than ``scattering-matrix formulation.'' We have offered a brief discussion of the advantages of our approach and the earlier works on transfer matrix in two and three dimensions in Ref. 33. We did not include a more detailed discussion in the present paper, because we intended to keep it focused on the specific problems we consider. We also believe that such a comparison should be given in a comprehensive review article on the subject.

Remark 4. The referee asks: ``In Eq. 12, is $p_0$ defined anywhere?''

Our response to Remark 4: The answer to the referee's question is ``yes.'' It is given in Eq. (81). This is perhaps the only instance in our manuscript where we missed to give the definition of a symbol before or right after it appears. This is however not a major shortcoming, because from Eq. (12) it is almost obvious what $p_0$ should mean. In the revised manuscript we will give its definition also below Eq. (12).

Remark 5. The referee writes: ``The paper defines many quantities without explaining their physical meaning. To facilitate the reading, perhaps the authors can add explanations. For example, consider adding after the definition in Eq. (26) that it is an expansion of the identity operator in momentum space with weights w(p).''

Our response to Remark 5: Frankly, we do not subscribe to the assertion that every mathematical quantity entering the solution of a physics problem must be given a physical explanation or interpretation. What is necessary and often missing in the works of mathematical physicists is that they do not even give an explanation or interpretation of the final result of a mathematical analysis which aims to deal with a physics problem. This does not apply to our work, we use a formalism to solve a class of physically relevant scattering problems and present a clear interpretation of our final results, explaining the physical effect of the presence of an exceptional point on the behavior of scattering amplitude. It is true that offering physical explanation for intermediate mathematical results may be helpful. But in our opinion it is not absolutely necessary. Depending on the problem at hand, it can make the analysis of the problem too lengthy and divert the attention of the reader. For example, the referee suggests that we consider interpreting Eq.~(26), which reads
\[\widehat\varpi:=\varpi(\hat p)=
\int_{-\infty}^\infty dp\,\varpi(p)|p\rangle\langle p|,~~~~~~~~~~~~~~(26)\]
as ``an expansion of the identity operator in momentum space with weights w(p).'' This is actually not how we wish the reader to view this equation. Eq. (26) defines $\widehat\varpi$ as the value of the function $\varpi$ at the operator $\hat p$ and then reminds the reader how one defines functions of a Hermitian operator in terms of its spectral resolution. This is basic knowledge of undergraduate QM, which in our opinion needs no explanation or interpretation in a research paper on quantum scattering. For example, when we give the following formula for the evolution operator of a free particle of mass $m$ in one dimension,
\[\exp\left[-\frac{i(t-t_0)\hat p^2}{2m\hbar}\right]:=\int_{-\infty}^\infty dp\,
\exp\left(-\frac{i(t-t_0)p^2}{2m\hbar}\right)|p\rangle\langle p|, \]
we do not wish to interpret it as ``an expansion of the identity operator in momentum space with weights $\exp(-\frac{i(t-t_0)p^2}{2m\hbar})$.'' This is simply because this quantity is not the identity operator (except for $t=t_0$.) Before giving the above relation for the evolution operator, we need to give the definition of the functions of a Hermitian operator in terms of their spectral resolution. Then when we give this relation, the student will take it as a specific example of what a function of a Hermitian operator means. In our opinion, for a student who has exposure to basic operator theory of elementary QM, Eq.(26) will require no interpretation; it is just another example of what the function of a Hermitian operator means.

Remark 6. The referee asks the following questions: 6.a)``Is the number of real eigenvalues of $H_{\rm hat}$ (on page 8) equal to the number of bound states in the waveguide?'' 6b) ``Do the complex-conjugate pairs correspond to the continuum of unbounded solutions?'' 6.c) ``What is the physical significance of their imaginary components? (penetration depths? lifetimes?)''

Our response to Remark 6: As we have tried to make it clear from the beginning the effective Hamiltonian $\widehat{\mathbf{H}}(x)$ has been introduced as a tool that allows us to determine the transfer matrix of the system. From his/her questions, it seems that the referee confuses this effective Hamiltonian with the Hamiltonian, $\mathbf{p}^2+v(\hat x,\hat y)$, that defines the scattering problem through the Schr\"odinger equation~(2).
Our answer to the above questions are as follows: 6.a) For the waveguide system, the number of real eigenvalues of $\widehat{\mathbf{H}}(x)$ is given by Eq. (124). It is clear from this relation that this number depends on the incident wave number. Therefore, even for cases where the waveguide has no bound states (i.e., $\mathscr{V}_0\geq 0$) the effective Hamiltonian $\widehat{\mathbf{H}}(x))$ can have real eigenvalues. Therefore, the answer to 6.a) is ``No.'' 6.b) and 6.c) Again for the same reason the answer is ``No.'' The complex eigenvalues always exist irrespectively of whether the guide has a bound state or not. In the generic case the real eigenvalues are also present and these also contribute to the scattering phenomenon (behavior of scattering states.) We have provided a detailed discussion of the contribution of real and complex eigenvalues and the corresponding eigenvectors of $\widehat{\mathbf{H}}(x)$ to the scattering amplitudes of the waveguide in Sec.~5. See the paragraphs containing Eqs. (124) -- (145). The referee seems to have ignored this discussion and instead tried to adopt the standard interpretation of real and complex eigenvalues of complex Hamiltonians modeling infinite waveguides with active regions to gain a qualitative understanding of those of the effective Hamiltonian $\widehat{\mathbf{H}}(x)$. This is simply not warranted.

Remark 7. For the waveguide system of Sec. 5, the referee suggests us to plot ``the eigenvalues and eigenfunctions, showing that the spectrum contains exceptional points'' and ``the dependence of scattering amplitudes on the geometrical or material parameters.'' He also write: ``Any plot of this sort and its discussion could have helped me appreciate the findings.''

Our response to Remark 7: We can certainly generate and add such plots to the paper. In the original manuscript, we did not do so because we were not sure how helpful they would be. For example, the eigenvalues are given by Eq. (112). So they are either real or imaginary and plotting them means plotting a bunch of square root functions. The same is true of eigenfunctions which are given by Eq. (92). Therefore plotting them means plotting a bunch of sine functions.

Remark 8. The referee suggests that we state our results on the dependence of the transmission amplitude on the length of the waveguide at an exceptional point in the abstract and introduction and that we should provide a graphical demonstration of this result.

Our response to Remark 8: We can do this in the revised version but again we have reservations on the real impact of the plots of transmission amplitude, for we give explicit analytic formulas which includes simple functions. For example, see Eq. (138). If we wish to plot this quantity we should fix all the variables except the length $a$. To demonstrate what happens at an exceptional point we should confine our attention to the case where only the term $n=1$ is present in (138). Plotting the real and imaginary parts of this quantity as a function of $a$ when we are at the exceptional point will give horizontal lines and when we are aways from the exceptional point we get sums of a sine and cosine functions. In preparing the revised manuscript, we will generate these graphs and try to see if their inclusion can improve the presentation of our results.

Remark 9. The referee asks us to elaborate a comment we make in the second paragraph of conclusions, namely ``In particular, for an empty waveguide, we have shown that at the exceptional wavenumbers, where H develops an exceptional point, the transmitted wave includes a term that is independent of the length of the waveguide.'' (S)he then writes: ``Does this refer to the same point as my previous remark or is this a different result? Perhaps the authors could plot the contribution to the transmission that is independent and the one that depends on the length, and show how they relate to each other?''

Our response to Remark 9: This result follows from Eq. (138) when the sum includes more than one term with the first representing an exceptional point. In the revised manuscript we also provide a discussion of this point in the paragraph containing Eq. (139). Regarding plotting the length-dependent and length-independent contributions to the transmission amplitude, we will consider their inclusion in the revised manuscript after we generate them and make sure that they can be of help for some readers.

Remark 10. The referee writes: ``The authors say in the concluding paragraph that the addition of loss or gain would make the EPs disappear. Could they explain why this happens?''

Our response to Remark 10: We answer this question in the last paragraph of Sec. 5. As we discussed in great detail in Sec. 3, an exceptional point corresponds to the case where $w_n=0$. In the last paragraph of Sec. 5, we provide an argument showing that when the guide includes active or lossy material $w_n\neq 0$. Therefore, no exceptional point can exist.