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

Spectra for the trigonometric (Et) and hyperbolic (Eh) system for M = 2, K = 1 and ζ = 25, ρ = 0 as a function of the field parameter η or topological index κ.

Konrad Schatz, Bretislav Friedrich, Simon Becker, Burkhard Schmidt

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 Schrödinger 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.

Phys. Rev. A 97 (5), 053417 (2018)