Fredholm determinants, continued fractions, Jost and Evans functions for a Jacobi matrix associated with the 2D-Euler equations
Abstract
For a second order difference equation that arises in the study of stability of unidirectional (generalized Kolmogorov) flows for the Euler equations of ideal fluids on the two dimensional torus, we relate the following five functions of the spectral parameter: the Fredholm determinants of the Birman-Schwinger operator pencils associated with the second order equation and the equivalent system of the first order equations; the Jost function constructed by means of the Jost solutions of the second order equation; the Evans function constructed by means of the matrix valued Jost solutions of the first order system, and, finally, to backward and forward continued fractions associated with the second order difference equation. We prove that all five functions are equal, and that their zeros are the discrete eigenvalues of the second order difference equation. We use this to improve an instability result for a generalization of the Kolmogorov (unidirectional) flow for the Euler equation on the 2D torus.
Keywords
Cite
@article{arxiv.2401.14037,
title = {Fredholm determinants, continued fractions, Jost and Evans functions for a Jacobi matrix associated with the 2D-Euler equations},
author = {Yuri Latushkin and Shibi Vasudevan},
journal= {arXiv preprint arXiv:2401.14037},
year = {2024}
}
Comments
Results in this paper improve and extend the results of arXiv:1901.01367v3 on the instability of unidirectional flows