Spectral flow, Brouwer degree and Hill's determinant formula

Alessandro Portaluri, Li Wu

Research output: Contribution to journalArticlepeer-review

Abstract

In 2005 a new topological invariant defined in terms of the Brouwer degree of a determinant map, was introduced by Musso, Pejsachowicz and the first name author for counting the conjugate points along a semi-Riemannian geodesic. This invariant was defined in terms of a suspension of a complexified family of linear second order Dirichlet boundary value problems. In this paper, starting from this result, we generalize this invariant to a general self-adjoint Morse-Sturm system and we prove a new spectral flow formula. Finally we discuss the relation between this spectral flow formula and the Hill's determinant formula and we apply this invariant for detecting instability of periodic orbits of a Hamiltonian system.

Original languageEnglish (US)
Pages (from-to)7253-7286
Number of pages34
JournalJournal of Differential Equations
Volume269
Issue number9
DOIs
StatePublished - Oct 15 2020

Keywords

  • Brouwer degree
  • Elliptic boundary value problems
  • Hill's determinant formula
  • Spectral flow
  • Trace formula

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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