Dyadic Resolvent Representations of Self-Adjoint Operators: Propagator Expansions, Spectral Measures, and Zeta Functions

What the math gives to ML

The paper provides an exact multiscale representation of a self-adjoint resolvent using unitary propagator samples at dyadic times. This can become a spectrally controlled neural layer: instead of applying a single polynomial graph filter, the model aggregates features after propagations at times 1, 1/2, 1/4, and so on, producing a rational spectral filter with explicit scale weights. The construction is most directly transferable to graph neural networks and state-space models whose latent dynamics are generated by symmetric or approximately skew-symmetric operators. A finite truncation with approximate propagators gives a practical architecture that can be tested against Chebyshev and standard polynomial graph filters at matched sparse-matrix cost.