Rotational symmetry and degeneracy: a cotangent-perturbed rigid rotator of unperturbed level multiplicity

TL;DR

This paper introduces an exactly solvable cotangent-perturbed rigid rotator model that maintains the same degeneracy as an unperturbed rotator, providing insights into rotational symmetry breaking and applications to anomalous rotational bands.

Contribution

The paper presents a novel exactly solvable model of a cotangent-perturbed rigid rotator that preserves degeneracy, using Romanovski polynomials and revealing new symmetry properties.

Findings

The energy spectrum is a linear combination of t(t+1) and 1/[t(t+1)+1/4] terms.

The model maintains (2t+1)-fold degeneracy similar to the unperturbed case.

Wave functions differ from spherical harmonics at small t but converge asymptotically.

Abstract

We predict level degeneracy of the rotational type in diatomic molecules described by means of a cotangent-hindered rigid rotator. The problem is shown to be exactly solvable in terms of non-classical Romanovski polynomials. The energies of such a system are linear combinations of t(t+1) and 1/[t(t+1)+1/4] terms with the non-negative integer principal quantum number t=n+|/bar{m}| being the sum of the degree n of the polynomials and the absolute value, |/bar{m}|, of the square root of the separation constant between the polar and azimuthal motions. The latter obeys, with respect to t, the same branching rule, |/bar{m}|=0,1,..., t, as does the magnetic quantum number with respect to the angular momentum, l, and, in this fashion, the t quantum number presents itself indistinguishable from l. In effect, the spectrum of the hindered rotator has the same (2t+1)-fold level multiplicity as the unperturbed one. For small t values, the wave functions and excitation energies of the perturbed rotator differ from the ordinary spherical harmonics, and the l(l+1) law, respectively, while approaching them asymptotically with increasing t. In this fashion the breaking of the rotational symmetry at the level of the representation functions is opaqued by the level degeneracy. The model provides a tool for the description of rotational bands with anomalously large gaps between the ground state and its first excitation.