Fredholm properties of nonlocal differential operators via spectral flow

TL;DR

This paper establishes Fredholm properties for nonlocal differential operators, introduces a spectral flow method to compute indices, and demonstrates applications in shock wave existence and bifurcation analysis.

Contribution

It introduces a generalized spectral flow approach to analyze Fredholm properties and indices of nonlocal differential operators, with applications in nonlinear and linear problems.

Findings

Fredholm properties established for a class of nonlocal operators.

Spectral flow method for computing Fredholm indices.

Applications include shock wave existence and bifurcation analysis.

Abstract

We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear and a linear setting. We first prove the existence of small viscous shock waves in nonlocal conservation laws with small spatially localized source terms. We also show how our results can be used to study edge bifurcations in eigenvalue problems using Lyapunov-Schmidt reduction instead of a Gap Lemma.