Monodromy in prolate spheroidal harmonics

Abstract

We show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators on L2false(S2false) have a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog of the Laplace–Runge–Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect, we construct a classical integrable system that is the semiclassical limit of the quantum integrable system of commuting operators. We show that this is a generalized semitoric system with a nondegenerate focus–focus point, such that there is monodromy in the classical and the quantum systems.

Spheroidal Domains and Geometric Analysis in Euclidean Space

“…Spheroidal domains, usually studied in terms of quaternion analysis, are here reformulated in the geometric analysis of Euclidean space. Spherical domains and spherical harmonics are a limiting case of spheroidal domains and spheroidal harmonics [3][4][5].…”