The Harmonic Lagrange Top and the Confluent Heun Equation

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