The infinite dimensional Evans function

We introduce generalized operator valued Jost solutions of first order ill-posed differential equations on Hilbert spaces. We then construct an infinite dimensional Evans function for abstract differential equations as a 2-modified Fredholm determinant of the operator obtained by adding the values a...

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Bibliographic Details
Published in:Journal of functional analysis Vol. 268; no. 6; pp. 1509 - 1586
Main Authors: Latushkin, Yuri, Pogan, Alin
Format: Journal Article
Language:English
Published: Elsevier Inc 15.03.2015
Subjects:
Fredholm determinants
Jost functions
Linear stability
Traveling waves
secondary
Linear stability
Traveling waves
Fredholm determinants
Jost functions
primary
ISSN:0022-1236, 1096-0783
Online Access:Get full text
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Summary:We introduce generalized operator valued Jost solutions of first order ill-posed differential equations on Hilbert spaces. We then construct an infinite dimensional Evans function for abstract differential equations as a 2-modified Fredholm determinant of the operator obtained by adding the values at zero of the generalized operator valued Jost solutions. Next, we prove a formula that connects the 2-modified Evans determinant and the 2-modified determinant related to the Birman–Schwinger type operator associated to the ill-posed equation. Using this formula, we construct a holomorphic infinite dimensional Evans function for second order differential operators on infinite cylinders whose zeros are the eigenvalues of the differential operators.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2014.11.020