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Fredholm determinants, continued fractions, Jost and Evans functions for a Jacobi matrix associated with the 2D-Euler equations

Yuri Latushkin, Shibi Vasudevan

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.

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.
Paper Structure (2 sections, 3 theorems, 46 equations)

This paper contains 2 sections, 3 theorems, 46 equations.

Table of Contents

  1. Introduction and the main results
  2. Proofs of the main results

Key Result

Theorem 1.1

Assume condrho and $\lambda\notin[-2{\rm i}, 2 {\rm i}]$. The functions introduced in defK, defT, defcE, defF and defG are holomorphic in $\lambda$ and equal, where the last equality holds under the additional assumption $\operatorname{Re}(\lambda)>0$. Moreover, $\lambda$ is a discrete eigenvalue of eulerev1 if and only if it is a zero of each of the functions in bigfive.

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • proof
  • proof