Classical and Quantum Integrable Systems in $\wt{\gr{gl}}(2)^{+*}$ and Separation of Variables

TL;DR

This paper demonstrates the complete integrability and separability of classical and quantum systems derived from spectral invariants on rational coadjoint orbits of a loop algebra, using hyperellipsoidal coordinates.

Contribution

It extends separation of variables techniques to both classical and quantum integrable systems on spheres and ellipsoids within the framework of loop algebra coadjoint orbits.

Findings

Classical systems are integrated via separation of variables in hyperellipsoidal coordinates.

Quantum systems are shown to be completely integrable and separable in the same coordinates.

Modified invariants of order O(ħ^2) are necessary for self-adjointness on ellipsoids.

Abstract

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $\wt{\gr{gl}}^{+*}(2,{\bf R})$ are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e., by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order $\OO(\hbar^2)$ in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. For each case - in the ambient space ${\bf R}^{n}$, the sphere and the ellipsoid - the Schr\"odinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lam\'e type.