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arXiv · 2610.16317

Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator

Ian Marquette·O. Ogulcan Tuncer·I. Yurdusen
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Abstract

We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-$\tfrac12$ 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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Ian Marquette, O. Ogulcan Tuncer, I. Yurdusen. 2026-10-12. Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator. https://arxiv.org/abs/2610.16317

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