Quantum Integrable Systems arising from Separation of Variables on S3

TL;DR

This paper investigates quantum integrable systems derived from separation of variables on the 3-sphere, classifying eigenfunctions, spectra, and monodromy across six coordinate systems, revealing degenerations and spectral defects.

Contribution

It provides a comprehensive classification of quantum integrable systems on S3 from all separable coordinates, analyzing their eigenfunctions, spectra, and degenerations, including the discovery of quantum monodromy.

Findings

Eigenfunctions classified by discrete symmetries.

Joint spectra computed for each coordinate system.

Quantum monodromy identified in the prolate system.

Abstract

We study the family of quantum integrable systems that arise from separating the Schr\"odinger equation in all 6 separable orthogonal coordinates on the 3 sphere: ellipsoidal, prolate, oblate, Lam\'{e}, spherical and cylindrical. On the one hand each separating coordinate system gives rise to a quantum integrable system on S2 x S2, on the other hand it also leads to families of harmonic polynomials in R4. We show that separation in ellipsoidal coordinates yields a generalised Lam\'{e} equation - a Fuchsian ODE with 5 regular singular points. We seek polynomial solutions so that the eigenfunctions are analytic at all finite singularities. We classify eigenfunctions by their discrete symmetry and compute the joint spectrum for each symmetry class. The latter 5 separable coordinate systems are all degenerations of the ellipsoidal coordinates. We perform similar analyses on these systems and show how the ODEs degenerate in a fashion akin to their respective coordinates. For the prolate system we show that there exists a defect in the joint spectrum which prohibits a global assignment of quantum numbers: the system has quantum monodromy. This is a companion paper to our previous work where the respective classical systems were studied.