PT-symmetry, indefinite damping and dissipation-induced instabilities

Abstract

Citation: Kirillov, Oleg (2012) PT-symmetry, indefinite damping and dissipation-induced instabilities. Physics Letters A, 376 (15 Northumbria University has developed Northumbria Research Link (NRL) to enable users to access the University's research output. Copyright © and moral rights for items on NRL are retained by the individual author(s) and/or other copyright owners. Single copies of full items can be reproduced, displayed or performed, and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided the authors, title and full bibliographic details are given, as well as a hyperlink and/or URL to the original metadata page. The content must not be changed in any way. Full items must not be sold commercially in any format or medium without formal permission of the copyright holder. The full policy is available online: http://nrl.northumbria.ac.uk/policies.html This document may differ from the final, published version of the research and has been made available online in accordance with publisher policies. To read and/or cite from the published version of the research, please visit the publisher's website (a subscription may be required.)This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues.Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. With perfectly balanced gain and loss, dynamical systems with indefinite damping can obey the exact PT -symmetry being marginally stable with a pure imaginary spectrum. At an exceptional point where the symmetry is spontaneously broken, the stability is lost via passing through a non-semi-simple 1 : 1 resonance. In the parameter space of a general dissipative system, marginally stable PT -symmetric ones occupy singularities on the boundary of the asymptotic stability. To observe how the singular surface governs dissipation-induced destabilization of the PT -symmetric system when gain and loss are not matched, an extension of recent experiments with PT -symmetric LRC circuits is proposed.

The physics of exceptional points

“…Quite generally and in the spirit of PT -symmetry, its consideration is of course not restricted to quantum systems. Here, we mention electronic circuits where the connection of instabilities in PT -symmetric systems and EPs is investigated experimentally [47] and theoretically [48].…”

Stabilizing and destabilizing perturbations of PT -symmetric indefinitely damped systems

“…Expressing the components d 12 and d 22 of the matrix D from the conditions (2.6) where we assume that k 12 = 0 and then changing coordinates as x 1 = z 1 + iz 2 , x 2 =x 1 , x 3 =ẋ 1 , x 4 =ẋ 2 , where i = √ −1 and the overbar denotes complex conjugation, we transform equation (2.1) to iẋ = Hx, where For example, when k 11 = k 22 , we have PH T = HP, where P = diag(1, −1, −1, 1) [26], that is, in this particular case, H is a PT -symmetric non-Hermitian Hamiltonian [8,9] that contains, for example, the model considered by Schindler et al [6]. Is H in (2.11) PT -symmetric also when k 11 = k 22 ?…”

Feedback and PT Symmetry in a class of Active LCR Circuits

“…The proposal is first applied to an electronic circuit consisting of two LC oscillators coupled by a gyrator [3], [4], [13], [14], and symmetric negative and positive resistors [15], [16]. The use of a gyrator simplifies the analysis, and avoids the need for magnetic (mutual inductance) couplings, allowing prototypes to be constructed from standard devices [17].…”