Krein signature for instability of PT -symmetric states

Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT -symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple …