Krein Signature in Hamiltonian and $\mathcal{PT}$-symmetric Systems

We explain the concept of Krein signature in Hamiltonian and $\mathcal{PT}$-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.