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Index theory for singular Lagrangian systems and Bessel-type differential operators

Xijun Hu, Alessandro Portaluri, Li Wu

TL;DR

This work develops a comprehensive index theory for singular Lagrangian and Sturm–Liouville-type operators, introducing a concrete Lagrangian-intersection framework that yields a spectral-flow formula and a Morse index theorem in the presence of singular endpoints and varying domains. It provides explicit decompositions, trace maps, and a generalization of classical SL-operator results, with detailed treatment of Bessel-type operators and their Fredholm properties. The theory is then applied to asymptotic N-body dynamics, linking Morse indices to boundary data and central configurations, and yielding finiteness criteria for Morse indices tied to the Rellich phenomenon. Altogether, the results offer both a unifying abstract framework and concrete tools for analyzing singular differential operators in physics and celestial mechanics, including new insights into eigenvalue disappearance and Rellich-type effects.

Abstract

The aim of the present manuscript is to develop an index theory for singular Lagrangian systems, with a particular focus on the important class of singular operators given by Bessel type differential operators. The main motivation is to address several challenges posed by singular operators, which appear in a wide range of applications: celestial mechanics (for instance, perturbations in planetary motion), oscillatory systems with time dependent forcing, electromagnetism (such as wave equations in nonuniform media), and quantum mechanics (notably certain Schroedinger equations with periodic potentials). We pursue two principal objectives. First, we establish a spectral flow formula and a Morse Index Theorem for gap-continuous paths of singular Sturm Liouville operators. By means of these index formulas, we construct a Morse index theory for a broad class of Bessel type differential operators and apply it to a family of asymptotic solutions of the gravitational n body problem. Finally, our new index theory provides new insight into a phenomenon first observed by Rellich concerning the spectrum of one-parameter families of Sturm Liouville operators with varying domains.

Index theory for singular Lagrangian systems and Bessel-type differential operators

TL;DR

This work develops a comprehensive index theory for singular Lagrangian and Sturm–Liouville-type operators, introducing a concrete Lagrangian-intersection framework that yields a spectral-flow formula and a Morse index theorem in the presence of singular endpoints and varying domains. It provides explicit decompositions, trace maps, and a generalization of classical SL-operator results, with detailed treatment of Bessel-type operators and their Fredholm properties. The theory is then applied to asymptotic N-body dynamics, linking Morse indices to boundary data and central configurations, and yielding finiteness criteria for Morse indices tied to the Rellich phenomenon. Altogether, the results offer both a unifying abstract framework and concrete tools for analyzing singular differential operators in physics and celestial mechanics, including new insights into eigenvalue disappearance and Rellich-type effects.

Abstract

The aim of the present manuscript is to develop an index theory for singular Lagrangian systems, with a particular focus on the important class of singular operators given by Bessel type differential operators. The main motivation is to address several challenges posed by singular operators, which appear in a wide range of applications: celestial mechanics (for instance, perturbations in planetary motion), oscillatory systems with time dependent forcing, electromagnetism (such as wave equations in nonuniform media), and quantum mechanics (notably certain Schroedinger equations with periodic potentials). We pursue two principal objectives. First, we establish a spectral flow formula and a Morse Index Theorem for gap-continuous paths of singular Sturm Liouville operators. By means of these index formulas, we construct a Morse index theory for a broad class of Bessel type differential operators and apply it to a family of asymptotic solutions of the gravitational n body problem. Finally, our new index theory provides new insight into a phenomenon first observed by Rellich concerning the spectrum of one-parameter families of Sturm Liouville operators with varying domains.

Paper Structure

This paper contains 28 sections, 75 theorems, 327 equations.

Table of Contents

  1. Introduction
  2. Highlights and Main Results
  3. An abstract spectral flow formula
  4. Symplectic Hilbert spaces and a decomposition formula
  5. An abstract spectral flow formula
  6. An Abstract Spectral Flow Formula for SL-Operators
  7. A splitting formula for gap-continuous paths
  8. An explicit decomposition and a trace map
  9. The trace map
  10. Two classical examples
  11. The Morse Index Theorem for Singular Sturm-Liouville
  12. Preliminary Results on Lagrangian Intersection Theory
  13. Morse Index Theorem for General Boundary Conditions
  14. About a Rellich counterexample to a spectral flow formula
  15. Bessel-type differential operators
  16. ...and 13 more sections

Key Result

Theorem 1

Under conditions (H1)– (H2) and assuming that $s \mapsto \Lambda_s \in \mathop{\mathrm{\mathrm{Lag}}}\nolimits(W, \omega|_W)$ is gap-continuous, the following spectral flow formula holds: where $\rho^{\mathop{\mathrm{\mathcal{T}}}\nolimits}$ denotes the push-forward symplectic form.

Theorems & Definitions (181)

  • Example 1.1
  • Theorem 1
  • Remark 1.2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 2.2
  • ...and 171 more