skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability

Abstract

We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian is necessarily pseudo-Hermitian regard-less of whether it is diagonalizable or not. This result is different from Mostafazadeh’s, which requires the Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at exceptional points where the Hamiltonian is not diagonalizable. Our result implies that PT-symmetry breaking is equivalent to the onset of instabilities of pseudo-Hermitian systems, which was systematically studied by Krein et al. in 1950s. In particular, we show that the mechanism of PT-symmetry breaking is the resonance between two eigen modes with opposite signs of actions.

Authors:
ORCiD logo [1]; ORCiD logo [2];  [3]
  1. Beijing Jiao Tong Univ. (China)
  2. Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Univ. of Science and Technology of China, Hefei (China)
  3. Univ. of Science and Technology of China, Hefei (China)
Publication Date:
Research Org.:
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1651133
Grant/Contract Number:  
AC02-09CH11466
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 61; Journal Issue: 1; Journal ID: ISSN 0022-2488
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Zhang, Ruili, Qin, Hong, and Xiao, Jianyuan. PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability. United States: N. p., 2020. Web. https://doi.org/10.1063/1.5117211.
Zhang, Ruili, Qin, Hong, & Xiao, Jianyuan. PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability. United States. https://doi.org/10.1063/1.5117211
Zhang, Ruili, Qin, Hong, and Xiao, Jianyuan. Thu . "PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability". United States. https://doi.org/10.1063/1.5117211. https://www.osti.gov/servlets/purl/1651133.
@article{osti_1651133,
title = {PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability},
author = {Zhang, Ruili and Qin, Hong and Xiao, Jianyuan},
abstractNote = {We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian is necessarily pseudo-Hermitian regard-less of whether it is diagonalizable or not. This result is different from Mostafazadeh’s, which requires the Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at exceptional points where the Hamiltonian is not diagonalizable. Our result implies that PT-symmetry breaking is equivalent to the onset of instabilities of pseudo-Hermitian systems, which was systematically studied by Krein et al. in 1950s. In particular, we show that the mechanism of PT-symmetry breaking is the resonance between two eigen modes with opposite signs of actions.},
doi = {10.1063/1.5117211},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 61,
place = {United States},
year = {2020},
month = {1}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 3 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

On Dirac's New Method of Field Quantization
journal, July 1943


Complex Extension of Quantum Mechanics
journal, December 2002


Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries
journal, August 2002

  • Mostafazadeh, Ali
  • Journal of Mathematical Physics, Vol. 43, Issue 8
  • DOI: 10.1063/1.1489072

Statistical parity-time-symmetric lasing in an optical fibre network
journal, November 2017

  • Jahromi, Ali K.; Hassan, Absar U.; Christodoulides, Demetrios N.
  • Nature Communications, Vol. 8, Issue 1
  • DOI: 10.1038/s41467-017-00958-x

On the structure of the two-stream instability–complex G-Hamiltonian structure and Krein collisions between positive- and negative-action modes
journal, July 2016

  • Zhang, Ruili; Qin, Hong; Davidson, Ronald C.
  • Physics of Plasmas, Vol. 23, Issue 7
  • DOI: 10.1063/1.4954832

Nonreciprocal Light Propagation in a Silicon Photonic Circuit
journal, August 2011


Pseudo-Hermitian Representation of Quantum Mechanics
journal, November 2010

  • Mostafazadeh, Ali
  • International Journal of Geometric Methods in Modern Physics, Vol. 07, Issue 07
  • DOI: 10.1142/s0219887810004816

The energy spectrum of complex periodic potentials of the Kronig–Penney type
journal, November 1999


Thermonuclear future of the Tokamak magnetic confinement device
journal, October 1969


Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation
journal, February 2016


Enhanced sensitivity at higher-order exceptional points
journal, August 2017

  • Hodaei, Hossein; Hassan, Absar U.; Wittek, Steffen
  • Nature, Vol. 548, Issue 7666
  • DOI: 10.1038/nature23280

Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
journal, January 2002

  • Mostafazadeh, Ali
  • Journal of Mathematical Physics, Vol. 43, Issue 1
  • DOI: 10.1063/1.1418246

Kelvin-Helmholtz instability is the result of parity-time symmetry breaking
journal, March 2019

  • Qin, Hong; Zhang, Ruili; Glasser, Alexander S.
  • Physics of Plasmas, Vol. 26, Issue 3
  • DOI: 10.1063/1.5088498

Real Spectra in Non-Hermitian Hamiltonians Having P T Symmetry
journal, June 1998


Parity–time synthetic photonic lattices
journal, August 2012

  • Regensburger, Alois; Bersch, Christoph; Miri, Mohammad-Ali
  • Nature, Vol. 488, Issue 7410
  • DOI: 10.1038/nature11298

Parity–time-symmetric whispering-gallery microcavities
journal, April 2014

  • Peng, Bo; Özdemir, Şahin Kaya; Lei, Fuchuan
  • Nature Physics, Vol. 10, Issue 5
  • DOI: 10.1038/nphys2927

Beam Dynamics in P T Symmetric Optical Lattices
journal, March 2008


Visualization of Branch Points in P T -Symmetric Waveguides
journal, August 2008


Experimental study of active LRC circuits with PT symmetries
journal, October 2011