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Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians. I linear amplification

by L. F. Alves da Silva, R. A. Dourado, and M. H. Y. Moussa

This is not the current version.

Submission summary

As Contributors: Rodrigo Dourado · Miled Moussa
Preprint link: scipost_202110_00013v1
Date submitted: 2021-10-11 17:03
Submitted by: Moussa, Miled
Submitted to: SciPost Physics
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for a TD non-Hermitian Hamiltonian H(t). Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=Ξ⁻¹(t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivaton of the symmetry associated with a TD non-Hermitian linear Hamiltonian: a cavity field under linear amplification. The computed TD symmetry operator is then particularized for the equivalent TI linear Hamiltonian and its PT-symmetric restriction. In this TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 with k odd [J. Phys. A 35, L467 (2002)]. The results here derived together with those in the sequel, where we extend our analysis for a TD non-Hermitian quadratic Hamiltonian, enables us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission scipost_202110_00013v2 on 1 December 2021

Reports on this Submission

Anonymous Report 1 on 2021-11-12 (Invited Report)


This is the first attempt to extend two basic theorems on time-independent non-Hermitian operators to time-dependent Hamiltonian operators.


Lacks physical motivation. Its presentation may be substantially improved. The paper can made much shorter.


The current version does not meet the standards of the journal. But I think the authors can be given a chance to respond to the criticisms I have listed in my report. An improved version may qualify for publication in SciencePost and other more specialized journals.

Requested changes

See the attached PDF.


  • validity: ok
  • significance: ok
  • originality: good
  • clarity: ok
  • formatting: acceptable
  • grammar: below threshold

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Anonymous on 2021-11-17  [id 1949]

The paper first presents briefly a formulation of a time-dependent anti-linear symmetric operator. Then the paper describes in details the computation of the time-dependent non-Hermitian Hamiltonian of a cavity field.

I don't recommend its publication.
First of all, the paper is difficult to understand because of bad presentation. I didn't quite understand the derivation of the time-dependent non-linear symmetric operator because the derivation in Sec. III is not described in detail. Then the calculation for the cavity field in Secs. V and VI is too much in detail.

After all, I don't see what is achieved by the authors' method. The authors frequently emphasize that their formulation is consistent with results in other papers, but I don't see if the authors' formulation is the unique extension. As another point, I don't see if the calculation in Secs. V and VI for a physical model is only achieved by the authors' formulation. The bottom line is "so what?"