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On the Absolutely Continuous Spectrum of Generalized Indefinite Strings.

Jonathan Eckhardt1,2, Aleksey Kostenko2,3

  • 1Department of Mathematical Sciences, Loughborough University, Epinal Way, Loughborough, Leicestershire LE11 3TU UK.

Annales Henri Poincare
|November 1, 2021
PubMed
Summary

This study demonstrates the stability of absolutely continuous spectra for generalized indefinite strings under perturbations. This finding clarifies the spectral support for the conservative Camassa-Holm flow in its dispersive regime.

Keywords:
34B0737K15Primary 34L05Secondary 34L25

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Area of Science:

  • Mathematical Physics
  • Spectral Theory
  • Differential Equations

Background:

  • Generalized indefinite strings are crucial in various areas of mathematical physics.
  • Understanding the stability of their absolutely continuous spectra is essential for theoretical advancements.
  • Previous research has laid groundwork, but specific model behaviors require further investigation.

Purpose of the Study:

  • To investigate the absolutely continuous spectrum of generalized indefinite strings.
  • To establish the stability of these spectra under wide perturbations using a specific approach.
  • To determine the spectral support for the conservative Camassa-Holm flow in the dispersive regime.

Main Methods:

  • Employed the Deift and Killip approach for spectral analysis.
  • Investigated two model examples of generalized indefinite strings.
  • Analyzed perturbations to establish spectral stability.

Main Results:

  • Established the stability of absolutely continuous spectra for generalized indefinite strings under broad perturbations.
  • Demonstrated that the absolutely continuous spectrum of the isospectral problem for the conservative Camassa-Holm flow is supported on [1/4, ∞).
  • Provided rigorous mathematical evidence for spectral properties of these systems.

Conclusions:

  • The study confirms the robustness of absolutely continuous spectra in generalized indefinite strings.
  • The findings offer significant insights into the spectral characteristics of the Camassa-Holm flow.
  • This work contributes to a deeper understanding of spectral theory in mathematical physics.