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A. Chernyavsky and D.E. Pelinovsky

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Krein signature for instability of PT-symmetric states

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**Abstract:**

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

**Keywords**:

PT-symmetry, Krein signature, nonlinear Schrodinger equation, perturbation theory for linear operators.