The Harmonic Lagrange Top and the Confluent Heun Equation

TL;DR

This paper studies the harmonic Lagrange top, a classical integrable system with a quadratic potential, and links its quantum mechanics to the confluent Heun equation, providing a global geometric description and spectral formulas.

Contribution

It offers a global Poisson geometric framework for the harmonic Lagrange top and connects its quantum spectrum to the confluent Heun equation, including explicit spectral matrix formulas.

Findings

Identifies four topological types of the harmonic Lagrange top.

Derives a pentadiagonal matrix representation of the Hamiltonian.

Establishes the connection between the quantum spectrum and the confluent Heun equation.

Abstract

The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on $T^*S^3$. This global description naturally leads to a rational parametrisation of the set of critical values of the energy-momentum map. We show that there are 4 different topological types for generic parameter values. The quantum mechanics of the harmonic Lagrange top is described by the most general confluent Heun equation (also known as the generalised spheroidal wave equation). We derive formulas for an infinite pentadiagonal symmetric matrix representing the Hamiltonian from which the spectrum is computed.