Krein Signature in Hamiltonian and PT-Symmetric Systems

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