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

TL;DR

This paper develops an index theory for singular Lagrangian systems, especially Bessel-type differential operators, addressing challenges in various physical applications and establishing spectral flow and Morse Index theorems.

Contribution

It introduces a new index theory for singular Sturm-Liouville operators, including spectral flow formulas and Morse Index theorems, with applications to gravitational n-body problems.

Findings

Established spectral flow formula for singular operators

Constructed Morse index theory for Bessel-type operators

Provided new insights into spectrum variation phenomena

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.