The Krein signature, Krein eigenvalues, and the Krein Oscillation Theorem

In this paper the problem of locating eigenvalues of negative Krein signature is considered for operators of the form JL, where J is skew-symmetric with bounded inverse and L is self-adjoint. A finite-dimensional matrix, hereafter referred to as the Krein matrix, associated with the eigenvalue problem JLu = λu is constructed with the property that if the Krein matrix has a nontrivial kernel for some z0, then ± √ −z0 ∈ σ(JL). The eigenvalues of the Krein matrix, i.e., the Krein eigenvalues, are real meromorphic functions of the spectral parameter, and have the property that their derivative at a zero is directly related to the Krein signature of the eigenvalue. The Krein Oscillation Theorem relates the number of zeros of a Krein eigenvalue to the number of eigenvalues with negative Krein signature. Because the construction of the Krein matrix is functional analytic in nature, it can be used for problems posed in more than one space dimension. This feature is illustrated in an example for which the spectral stability of the dipole solution to the Gross-Pitaevski equation is considered.