Graphical Krein signature theory and Evans–Krein functions

Two concepts, evidently very different in nature, have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half-plane leading to dynamical instability under perturbation of the system, and (ii) the Evans function, an analytic function detecting the location of eigenvalues. One might expect these two concepts to be related, but unfortunately examples demonstrate that there is no way in general to deduce the Krein signature of an eigenvalue from the Evans function, for example by studying derivatives of the latter. The purpose of this paper is to recall and popularize a simple graphical interpretation of the Krein signature well-known in the spectral theory of polynomial operator pencils. Once established, this in-terpretation avoids altogether the need to view the Krein signature in terms of root subspaces and their relation to indefinite quadratic forms. To demonstrate the utility of this graphical interpretation of the Krein signature, we use it to define a simple generalization of the Evans function — the Evans-Krein function — that allows the calculation of Krein signatures in a way that is easy to incorporate int