Krein Signature in Hamiltonian and $\mathcal{PT}$-symmetric Systems

TL;DR

This paper explores the Krein signature in Hamiltonian and $\ ext{PT}$-symmetric systems using the Gross-Pitaevskii equation, revealing conditions for eigenvalue stability and instability relevant to Bose-Einstein condensates.

Contribution

It introduces a real-valued Krein quantity for the linearized Gross-Pitaevskii equation and analyzes eigenvalue stability criteria in $\ ext{PT}$-symmetric potentials.

Findings

Neutrally stable eigenvalues are nonzero and simple, indicating stability.

Multiple eigenvalues can split into unstable ones under perturbations.

Existence of two simple neutrally stable eigenvalues with opposite Krein signatures signals potential instability.

Abstract

We explain the concept of Krein signature in Hamiltonian and $\mathcal{PT}$-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 Krein quantity, which is nonzero if the eigenvalue is neutrally stable and simple and zero if the eigenvalue is unstable. If the neutrally stable eigenvalue is simple, it persists with respect to perturbations. However, if it is multiple, it may split into unstable eigenvalues under perturbations. A necessary condition for the onset of instability past the bifurcation point requires existence of two simple neutrally stable eigenvalues of opposite Krein signatures before the bifurcation point. This property is useful in the parameter continuations of neutrally stable eigenvalues of the linearized Gross-Pitaevskii equation.