Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator
Abstract
We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-1/2 degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular--spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series su(1,1) representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional j=0 sector, complete energy-shell multiplicities, and the effect of particle interchange.
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BibTeX
@misc{marquette2026nested,
title = {{Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator}},
author = {Ian Marquette and O. Ogulcan Tuncer and I. Yurdusen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16317},
eprint = {2610.16317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16317},
}
RIS
TY - UNPB
TI - Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator
AU - Ian Marquette
AU - O. Ogulcan Tuncer
AU - I. Yurdusen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16317
UR - https://arxiv.org/abs/2610.16317
AB - We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-1/2 degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular--spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series su(1,1) representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional j=0 sector, complete energy-shell multiplicities, and the effect of particle interchange.
ER -