Symmetric Tops Subject to Combined Electric Fields: Conditional Quasi-Solvability via the Quantum Hamilton-Jacobi Theory

TL;DR

This paper uses the Quantum Hamilton-Jacobi theory to analyze the conditional quasi-solvability of the symmetric top under electric fields, deriving exact solutions and revealing spectral patterns.

Contribution

It introduces a novel application of the QHJ approach to find exact solutions for the symmetric top in combined electric fields, extending previous methods.

Findings

Derived conditions for quasi-solvability of the system.

Obtained finite sets of exact analytic solutions.

Revealed spectral pattern similarities between related systems.

Abstract

We make use of the Quantum Hamilton-Jacobi (QHJ) theory to investigate conditional quasi-solvability of the quantum symmetric top subject to combined electric fields (symmetric top pendulum). We derive the conditions of quasi-solvability of the time-independent Schroedinger equation as well as the corresponding finite sets of exact analytic solutions. We do so for this prototypical trigonometric system as well as for its anti-isospectral hyperbolic counterpart. An examination of the algebraic and numerical spectra of these two systems reveals mutually closely related patterns. The QHJ approach allows to retrieve the closed-form solutions for the spherical and planar pendula and the Razavy system that had been obtained in our earlier work via Supersymmetric Quantum Mechanics as well as to find a cornucopia of additional exact analytic solutions.