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Spectral flow and variational bifurcation

J. Pejsachowicz

Abstract

We show that the principle "nonvanishing of spectral flow of the linearization along the trivial branch entails bifurcation of nontrivial solutions ", proved in \cite{FPR} for critical points of one parameter families of $C^2$ functionals with Fredholm Hessian, holds true for variational perturbations of paths of unbounded self-adjoint Fredholm operators with a fixed domain.

Spectral flow and variational bifurcation

Abstract

We show that the principle "nonvanishing of spectral flow of the linearization along the trivial branch entails bifurcation of nontrivial solutions ", proved in \cite{FPR} for critical points of one parameter families of functionals with Fredholm Hessian, holds true for variational perturbations of paths of unbounded self-adjoint Fredholm operators with a fixed domain.
Paper Structure (14 sections, 13 theorems, 42 equations)

This paper contains 14 sections, 13 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Variational Bifurcation
  3. Symplectic Hilbert spaces
  4. Fredholm pairs
  5. Topology of the Fredholm Lagrangian Grassmannian
  6. Graphs of self-adjoint operators and Lagrangians
  7. Symplectic reduction
  8. Generalized Maslov Index and Spectral Flow
  9. Maslov index in finite dimensions
  10. The generalized Maslov index
  11. Spectral flow of a path of unbounded self-adjoint Fredholm operators.
  12. Proofs
  13. Proof of Theorem \ref{['thm:riduzione']}
  14. Proof of Theorem \ref{['thm:teoremadibiforca']}

Key Result

Theorem 1.2

Let $f(\lambda,u)= A_\lambda u + F(\lambda,u)$ be a variational perturbation of an admissible path ${\mathcal{A}}$ in $\mathcal{L}(H_2;H)$ verifying pert and grad.

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • ...and 19 more