Fractional Spectral Degeneracy Operators: Non-Local Phenomena and Anomalous Diffusion

We develop a unified and mathematically rigorous framework for Fractional Spectral Degeneracy Operators (FSDOs), a broad class of anisotropic non-local operators that combine fractional diffusion with spatially dependent degeneracy of variable strength. This formulation generalizes classical Spectral Degeneracy Operators by allowing degeneracy exponents θ_i ∈ (0, 2), thereby capturing a continuum of diffusion regimes ranging from mildly singular behavior to near-critical ultra-degeneracy. Motivated by applications in anomalous transport, intermittent turbulence, and heterogeneous or fractal media, we introduce weighted fractional Sobolev spaces tailored to the anisotropic metric generated by the degeneracy. Within this setting, we establish fractional Hardy and Poincaré inequalities that guarantee coercivity and control of the associated bilinear forms. Building on these foundations, we prove essential self-adjointness, compact resolvent, and a complete spectral decomposition for FSDOs. A detailed heat kernel parametrix is constructed using anisotropic pseudo-differential calculus, yielding sharp small-time asymptotics and, through a Tauberian argument, a fractional Weyl law whose exponent depends explicitly on the dominant degeneracy direction. We further obtain Bessel-type expansions for eigenfunctions near the singular locus and derive fractional Landau inequalities that encode an uncertainty principle adapted to the weighted fractional geometry. As an application, we introduce Fractional SDO-Nets, a class of neural operators whose layers incorporate the inverse FSDO. These architectures inherit stability, non-locality, and anisotropic scaling from the underlying operator and provide a principled mechanism for learning fractional and degenerate diffusion phenomena from data.