Spectral Analysis of the Helical Vorticity Operator in a Bounded Spherical Domain

We present a rigorous spectral analysis of the first-order helical vorticity operator Dˆ =⃗σ · (∇×⃗ ) defined on the Hilbert space of divergence-free (transverse) vector fields within abounded spherical domain of finite radius R0. Subject to strict Dirichlet boundary conditions, we formalize the underlying functional space V and analytically investigate thesymmetric, topological, and eigenvalue properties of the configuration. We demonstratethat on the space of transverse modes, the square of the helical operator maps formally ontothe spatial Laplacian, Dˆ2 = −∇⃗ 2, providing an explicit tensor proof of the vanishing antisymmetric Clifford sector. Utilizing the framework of Vector Spherical Harmonics (VSH),we achieve complete separation of variables and reduce the system to spherical Bessel differential equations. Invoking the Rellich-Kondrachov compactness theorem, we prove theexistence of a purely discrete spectrum. The eigenvalue distribution reveals exact, symmetric paired chiral branches ±λn,l corresponding to the non-zero roots of higher-order sphericalBessel functions. This spectral topology establishes a mathematically controlled foundationfor stable chiral states without invoking non-local axiomatic postulates. A tabulated presentation of the lower multipole eigenvalue energy levels (j = 3/2, 5/2, 7/2) alongside a minimalexecutable verification script is attached to validate the framework.