The Evans function as a lower bound on the spectral distance function

TL;DR

This paper introduces a normalized Evans function that provides a lower bound on the spectral distance, aiding in stability analysis of differential operators by linking the Evans function's magnitude to the resolvent set.

Contribution

It proposes a natural normalization of the Evans function that directly relates its magnitude to the spectral distance, enhancing spectral stability analysis tools.

Findings

Normalized Evans function bounds the spectral distance from the spectrum.

Numerical experiments demonstrate the method on various operators.

The approach applies to boundary value problems in stability analysis.

Abstract

The Evans function is an analytic function that encodes information about the intersection of certain subspaces in ODE boundary value problems. As such it is a useful tool for computing the spectrum of boundary value problems arising in the stability of coherent structures. In typical applications one is interested in the roots of the Evans function, but the overall normalization is somewhat arbitrary. We present a natural normalization of the Evans function on compact domains such that the magnitude of the Evans function provides a lower bound on the distance to the nearest point in the spectrum. In other words the magnitude of the Evans function at a point in the resolvent set implies that a ball about the point in question lies in the resolvent set. Thus, when appropriately normalized, not only does the Evans function $E(\lambda)$ vanish if and only if $\lambda$ lies in the spectrum of the operator in question, but a non-zero value for the Evans function guarantees that a disk of radius $|E(\lambda^*)|$ about the point $\lambda^*$ lies in the resolvent set. We present some calculations for some common sets of boundary conditions on a compact interval, and present some numerical experiments for 2nd and 4th order self-adjoint operators and for a linearized modified Korteweg-De Vries equation.