  Krein quantity is introduced for isolated neutrally stable eigenvalues
associated with the stationary states in the $\mathcal{PT}$-symmetric nonlinear
Schr\"{o}dinger 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
eigenvalues before the coalescence point to have opposite Krein signatures. The
theory is illustrated with several numerical examples motivated by recent
publications in physics literature.
