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Index theory for non-compact quantum graphs

Daniele Garrisi, Alessandro Portaluri, Li Wu

Abstract

We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges. These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems.

Index theory for non-compact quantum graphs

Abstract

We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges. These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems.

Paper Structure

This paper contains 19 sections, 25 theorems, 175 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Spectral Flow Formula on (non-compact) graphs.
  3. The Morse Index Theorem on (non-compact) graphs.
  4. Variational framework on graphs
  5. Gelfand-Robbin quotient and Cauchy data spaces
  6. A spectral flow formula in the Cauchy data spaces
  7. The Morse index theorem on (non)compact graphs
  8. A spectral flow formula on graphs
  9. The Morse index theorem on graphs
  10. Difference of Morse index
  11. Some applications to star graphs
  12. A reduction formula for the union of two star graphs
  13. The Morse index theorem for a compact star graph
  14. The Morse index theorem for a two-star graph
  15. Spectral flow and Maslov index
  16. ...and 4 more sections

Key Result

Theorem 1

Under the above notation, the following equality holds where $\iota$ denotes the triple index. (Cfr. Appendix sec:Maslov and references therein).

Figures (3)

  • Figure 1: A star graph with 5 leaves and outward orientation.
  • Figure 2: Directed star graph with $m=3$ leaves.
  • Figure 3: A two-star graph as which is the union of two star graphs having 3 and 4 leaf vertices, respectively

Theorems & Definitions (74)

  • Theorem 1
  • Example 1.1
  • Remark 2.3
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.6
  • proof
  • Remark 3.7
  • ...and 64 more