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

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.