$2$-modified characteristic Fredholm determinants, Hill's method, and the periodic Evans function of Gardne

TL;DR

This paper establishes a detailed connection between various Evans functions and Fredholm determinants for periodic differential operators, extending previous results to a broader class of operators and clarifying the spectral correspondence.

Contribution

It provides an explicit link between the generalized Birman--Schwinger-type Evans function and the standard Evans function for a wide family of operators, generalizing prior scalar and vector cases.

Findings

Connected the generalized Evans function with the standard Evans function.

Extended results of Gesztesy--Makarov and Gardner to broader operator classes.

Confirmed the zeros of Evans functions match the periodic eigenvalues in location and multiplicity.

Abstract

Using the relation established by Johnson--Zumbrun between Hill's method of aproximating spectra of periodic-coefficient ordinary differential operators and a generalized periodic Evans function given by the $2$-modified characteristic Fredholm determinant of an associated Birman--Schwinger system, together with a Volterra integral computation introduced by Gesztesy--Makarov, we give an explicit connection between the generalized Birman--Schwinger-type periodic Evans function and the standard Jost function-type periodic Evans function defined by Gardner in terms of the fundamental solution of the eigenvalue equation written as a first-order system. This extends to a large family of operators the results of Gesztesy--Makarov for scalar Schr\"odinger operators and of Gardner for vector-valued second-order elliptic operators, in particular recovering by independent argument the fundamental result of Gardner that the zeros of the Evans function agree in location and (algebraic) multiplicity with the periodic eigenvalues of the associated operator