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When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems itis especially important to locate not only the unstable eigenvalues (i.e.,those with positive real part), but also those which are purely imaginary buthave negative Krein signature. These latter eigenvalues have the property thatthey can become unstable upon collision with other purely imaginaryeigenvalues, i.e., they are a necessary building block in the mechanism leadingto the so-called Hamiltonian-Hopf bifurcation. In this paper we review ageneral theory for constructing a meromorphic matrix-valued function, theso-called Krein matrix, which has the property of not only locating theunstable eigenvalues, but also those with negative Krein signature. Theseeigenvalues are realized as zeros of the determinant. The resulting finitedimensional problem obtained by setting the determinant of the Krein matrix tozero presents a valuable simplification. In this paper the usefulness of thetechnique is illustrated through prototypical examples of spectral analysis ofstates that have arisen in recent experimental and theoretical studies ofatomic Bose-Einstein condensates. In particular, we consider one-dimensionalsettings (the cigar trap) possessing real-valued multi-dark-soliton solutions,and two-dimensional settings (the pancake trap) admitting complex multi-vortexstationary waveforms.

The Krein Matrix: General Theory and Concrete Applications in Atomic Bose-Einstein Condensates