A. Chernyavsky, P.G. Kevrekidis, D.E. Pelinovsky
Krein signature in Hamiltonian and PT-symmetric systems
Parity-time Symmetry and Its Applications
Editors: D. Christodoulides and J. Yang
Springer Tracts in Modern Physics 280 (Springer, New York) 465-491 (2019)
Abstract:
We explain the concept of Krein signature in Hamiltonian and 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.
Keywords:
Hamiltonian systems, PT-symmetry, Krein signature, nonlinear Schrodinger equation,
perturbation theory for linear operators.