Characteristic determinants for a second order difference equation on the half-line arising in hydrodynamics

TL;DR

This paper investigates the spectral properties of a second order difference operator with complex potential on the half-line, using advanced analytical tools, and applies findings to fluid flow stability analysis.

Contribution

It introduces a novel approach to analyze the point spectrum of difference operators via Fredholm determinants and applies it to fluid dynamics stability problems.

Findings

Spectral analysis of the difference operator using Fredholm determinants.

Application to instability analysis of generalized Kolmogorov flow.

Insights into fluid flow stability on the two-dimensional torus.

Abstract

We study the point spectrum of a second order difference operator with complex potential on the half-line via Fredholm determinants of the corresponding Birman-Schwinger operator pencils, the Evans and the Jost functions. An application is given to instability of a generalization of the Kolmogorov flow for the Euler equation of ideal fluid on the two dimensional torus.