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Krein Signature in Hamiltonian and \(\mathbb {PT}\)-Symmetric Systems

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Parity-time Symmetry and Its Applications

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 280))

Abstract

We explain the concept of Krein signature in Hamiltonian and \(\mathbb {PT}\)-symmetric systems on the case study of the one-dimensional Gross–Pitaevskii equation with a real harmonic potential and an imaginary linear potential. These potentials correspond to the magnetic trap, and a linear gain/loss in the mean-field model of cigar-shaped Bose–Einstein condensates. For the linearized Gross–Pitaevskii equation, we introduce the real-valued Krein quantity, which is nonzero if the eigenvalue is neutrally stable and simple and zero if the eigenvalue is unstable. If the neutrally stable eigenvalue is simple, it persists with respect to perturbations. However, if it is multiple, it may split into unstable eigenvalues under perturbations. A necessary condition for the onset of instability past the bifurcation point requires existence of two simple neutrally stable eigenvalues of opposite Krein signatures before the bifurcation point. This property is useful in the parameter continuations of neutrally stable eigenvalues of the linearized Gross–Pitaevskii equation.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, South Africa

    A. Chernyavsky

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA

    P. G. Kevrekidis

  3. Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

    D. E. Pelinovsky

Authors
  1. A. Chernyavsky
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  2. P. G. Kevrekidis
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  3. D. E. Pelinovsky
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Correspondence to D. E. Pelinovsky .

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Editors and Affiliations

  1. College of Optics and Photonics, University of Central Florida, Orlando, FL, USA

    Demetrios Christodoulides

  2. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

    Jianke Yang

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Chernyavsky, A., Kevrekidis, P.G., Pelinovsky, D.E. (2018). Krein Signature in Hamiltonian and \(\mathbb {PT}\)-Symmetric Systems. In: Christodoulides, D., Yang, J. (eds) Parity-time Symmetry and Its Applications. Springer Tracts in Modern Physics, vol 280. Springer, Singapore. https://doi.org/10.1007/978-981-13-1247-2_16

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  • DOI: https://doi.org/10.1007/978-981-13-1247-2_16

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