Supersymmetry and eigensurface topology of the spherical quantum pendulum

We undertook a mutually complementary analytic and computational study of the full-fledged spherical (3D) quantum rotor subject to combined orienting and aligning interactions characterized, respectively, by dimensionless parameters $eta$ and $zeta$. By making use of supersymmetric quantum mechanics (SUSY QM), we found two sets of conditions under which the problem of a spherical quantum pendulum becomes analytically solvable. These conditions coincide with the loci $zeta=frac{eta^2}{4k^2}$ of the intersections of the eigenenergy surfaces spanned by the $eta$ and $zeta$ parameters. The integer topological index $k$ is independent of the eigenstate and thus of the projection quantum number $m$. These findings have repercussions for rotational spectra and dynamics of molecules subject to combined permanent and induced dipole interactions.