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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_00035v2
Date submitted: 2022-01-14 16:01
Submitted by: Mostafazadeh, Ali
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Quantum Physics
Approach: Theoretical

Abstract

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


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 1 on 2022-1-19 (Invited Report)

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.

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