Separation of variables for the classical and quantum Neumann model

TL;DR

This paper demonstrates that the separation of variables technique applies to both classical and quantum Neumann models, simplifying their analysis by linearizing flows and reducing quantum equations to generalized Lamé forms.

Contribution

It extends the separation of variables method to the Neumann model in both classical and quantum contexts, revealing new analytical simplifications.

Findings

Classical flow linearized on the Jacobian of the spectral curve.

Quantum Schrödinger equation reduces to generalized Lamé equations.

Method provides a unified approach for classical and quantum Neumann models.

Abstract

The method of separation of variables is shown to apply to both the classical and quantum Neumann model. In the classical case this nicely yields the linearization of the flow on the Jacobian of the spectral curve. In the quantum case the Schr\"odinger equation separates into one--dimensional equations belonging to the class of generalized Lam\'e differential equations.