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Derivatives of the Evans function and (modified) Fredholm determinants for first order systems. (English) Zbl 1225.47131

Summary: The Evans function is a Wronskian type determinant used to detect point spectrum of differential operators obtained by linearizing PDEs about special solutions such as traveling waves, etc. This work is a sequel to the paper [F. Gesztesy, Y. Latushkin and K. Zumbrun, J. Math. Pures Appl. (9) 90, No. 2, 160–200 (2008; Zbl 1161.47058)], where the Evans and Jost functions for Schrödinger equations have been considered.
In the present work, we study the Evans function for the general case of linear ODE systems, and choose it to agree with the modified Fredholm determinant of the respective Birman-Schwinger type integral operator. The Evans function is thus the determinant of the matrix composed of the so-called generalized Jost solutions. These are the solutions of the homogeneous perturbed differential equation which are asymptotic to some reference solutions of the unperturbed equation.
One of the main results of the present paper is a formula for the derivative of the Evans function for the first order systems. Its proof uses a matrix composed of the newly introduced modified Jost solutions. These are the solutions of an inhomogeneous perturbed differential equation with the inhomogeneous term constructed by means of the above-mentioned generalized Jost solutions.

MSC:

47N20 Applications of operator theory to differential and integral equations
34A30 Linear ordinary differential equations and systems
34L05 General spectral theory of ordinary differential operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A55 Perturbation theory of linear operators

Citations:

Zbl 1161.47058
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References:

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