Graphical Krein Signature Theory and Evans-Krein Functions

Two concepts, very different in nature, have proved to be useful in analytical and numerical studies of spectral stability: (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. 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. This interpretation 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 into existing Evans function evaluation codes at virtually no additional computational cost. The graphical Krein signature also enables us to give elegant proofs of index theorems for linearized Hamiltonians in the finite dimensional setting: a general result implying as a corollary the generalized Vakhitov-Kolokolov criterion (or Grillakis-Shatah-Strauss criterion) and a count of real eigenvalues for linearized Hamiltonian systems in canonical form. These applications demonstrate how the simple graphical nature of the Krein signature may be easily exploited.

💡 Research Summary

The paper revisits two classical tools for spectral stability analysis—Krein signatures and Evans functions—and shows how a simple graphical interpretation of the former can be merged with the latter to create a new computational device, the Evans‑Krein function. Krein signatures indicate whether an eigenvalue on the imaginary axis of a Hamiltonian or other conservative system is structurally unstable, i.e., whether a small perturbation can push it into the right half‑plane and cause dynamical instability. Traditionally, the signature is defined via the relative orientation of invariant subspaces and indefinite quadratic forms, which makes its practical computation cumbersome, especially in infinite‑dimensional settings. The Evans function, by contrast, is an analytic scalar function whose zeros coincide with eigenvalues; it is widely used for numerical root‑finding but carries no information about the sign of the Krein signature.

The authors recall a well‑known graphical viewpoint from the theory of polynomial operator pencils. For a pencil (A(\lambda)=A_0+\lambda A_1+\dots+\lambda^k A_k), the eigenvalues are the roots of (\det A(\lambda)=0). Plotting the curves defined by (\det A(\lambda)=0) in the ((\lambda,\mu))‑plane (with (\mu) representing the spectral parameter) yields a family of branches that intersect the real (\lambda)‑axis. At each intersection the slope of the branch determines the Krein signature: a positive slope corresponds to a signature (+1), a negative slope to (-1). Thus the signature can be read off directly from the geometry of the graph, without any reference to root subspaces or indefinite forms.

To exploit this observation computationally, the paper introduces the Evans‑Krein function. Starting from a standard Evans function (D(\lambda)=\det M(\lambda)) (where (M) is a matrix built from stable/unstable manifolds), a small auxiliary parameter (\varepsilon) is added, forming (D(\lambda,\varepsilon)=\det