Potential-free $L^1$-estimates for positivity-preserving Riesz transform related to Schrödinger operator in dimension one

What the math gives to ML

The paper proves a dimension-one operator bound that is unusually robust: the positivity-preserving fractional resolvent V^a(-Delta+V)^(-a) has an L1 norm bounded independently of the nonnegative potential V. This suggests a neural sequence module that applies potential-weighted fractional diffusion or resolvent filtering, with the potential controlling local smoothing strength while the theorem supplies stability against arbitrary potential amplitudes. The most direct transfer is a learnable, nonnegative, lengthwise preconditioning layer for 1D signals, state-space inputs, or token channels, implemented with matrix-function approximations rather than explicit inversion. The theorem does not guarantee task accuracy, so the falsifiable benefit is improved stability or robustness at comparable compute, especially when the learned potential becomes highly nonuniform.