# Krein signature for instability of $\mathcal{PT}$-symmetric states

## TL;DR

This paper introduces the Krein quantity for analyzing the stability of $\\mathcal{PT}$-symmetric states in nonlinear Schrödinger equations, providing a criterion for eigenvalue bifurcations related to stability.

## Contribution

The paper develops a novel Krein quantity concept for $\\mathcal{PT}$-symmetric states and establishes a necessary condition for eigenvalue bifurcation leading to instability.

## Key findings

- Krein quantity is real and nonzero for simple eigenvalues.
- Bifurcation of unstable eigenvalues requires eigenvalues to have opposite Krein signatures.
- Numerical examples demonstrate the theory's applicability.

## Abstract

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.