Sharp Time-Decay Estimates for Fractional Heat Semigroups Associated with Polynomial Anharmonic Oscillators

What the math gives to ML

The paper provides a principled family of anisotropic, confining diffusion operators whose dynamics smooth in frequency while suppressing activity in spatial regions selected by a polynomial potential. The transferable asset is the positive self-adjoint Hamiltonian structure: fractional semigroup filters remain stable, have tunable short-time anisotropic smoothing, and contract exponentially at a rate determined by the lowest eigenvalue. A practical neural-network translation is a learnable phase-space diffusion layer, implemented with alternating Fourier-domain kinetic filtering and pointwise spatial potential damping, optionally augmented by a fractional spectral filter. This is most promising as a replacement for ad hoc smoothing or normalization in vision, spatiotemporal, and neural-operator models.