Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals

Beyn W-J, Latushkin Y, Rottmann-Matthes J (2014)
Integral Equations and Operator Theory 78(2): 155-211.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
Abstract / Bemerkung
Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh' theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are well-controlled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schrodinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHugh-Nagumo system. We also relate the contour method to the well-known Evans function and show that our approach provides an alternative to evaluating and computing its zeros.
Erscheinungsjahr
Zeitschriftentitel
Integral Equations and Operator Theory
Band
78
Ausgabe
2
Seite(n)
155-211
ISSN
PUB-ID

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Beyn W-J, Latushkin Y, Rottmann-Matthes J. Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals. Integral Equations and Operator Theory. 2014;78(2):155-211.
Beyn, W. - J., Latushkin, Y., & Rottmann-Matthes, J. (2014). Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals. Integral Equations and Operator Theory, 78(2), 155-211. doi:10.1007/s00020-013-2117-6
Beyn, W. - J., Latushkin, Y., and Rottmann-Matthes, J. (2014). Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals. Integral Equations and Operator Theory 78, 155-211.
Beyn, W.-J., Latushkin, Y., & Rottmann-Matthes, J., 2014. Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals. Integral Equations and Operator Theory, 78(2), p 155-211.
W.-J. Beyn, Y. Latushkin, and J. Rottmann-Matthes, “Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals”, Integral Equations and Operator Theory, vol. 78, 2014, pp. 155-211.
Beyn, W.-J., Latushkin, Y., Rottmann-Matthes, J.: Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals. Integral Equations and Operator Theory. 78, 155-211 (2014).
Beyn, Wolf-Jürgen, Latushkin, Yuri, and Rottmann-Matthes, Jens. “Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals”. Integral Equations and Operator Theory 78.2 (2014): 155-211.