Sharp Broken-Power Lorentz Estimates for Fractional Powers of Radial Schrödinger Operators with Inverse-Square Asymptotics

What the math gives to ML

The paper provides an explicit nonhomogeneous fractional-integration kernel whose behavior changes between the origin and infinity through a positive ground state U. This is transferable as a fixed or learnable multiscale mixing operator for token, image, or mesh features: unlike translation-invariant convolution, its interaction strength depends jointly on pairwise distance and each endpoint's radial scale. The clean kernel formula supplies a concrete positional bias, while the sharp broken-power Lorentz theory suggests how to stress-test stability under highly nonuniform input and output amplitudes. The most practical first experiment is a ground-state-modulated attention layer with two-regime radial exponents, compared against ordinary relative-position attention at equal parameter count and FLOPs.