Dynamic and geometric shifts in wave scattering

Since Berry's pioneering 1984 work, the separation of<i>geometric</i>and<i>dynamic</i>contributions in the<i>phase</i>of an evolving wave has become fundamental in physics, underpinning diverse phenomena in quantum mechanics, optics, and condensed matter. Here we extend this geometric-dynamic decomposition from the wave-evolution phase to a distinct class of<i>wave scattering</i>problems, where observables (such as frequency, momentum, or position) experience<i>shifts in their expectation values</i>between the input and output wave states. We describe this class of problems using a unitary scattering matrix and the associated<i>generalized Wigner-Smith operator</i>(GWSO), which involves gradients of the scattering matrix with respect to conjugate variables (time, position, or momentum, respectively). We show that both the GWSO and the resulting expectation-values shifts admit gauge-invariant decompositions into dynamic and geometric parts, related respectively to gradients of the<i>eigenvalues</i>and<i>eigenvectors</i>of the scattering matrix. We illustrate this general theory through a series of examples, including frequency shifts in polarized-light transmission through a time-varying waveplate (linked to the Pancharatnam-Berry phase), momentum shifts at spatially varying metasurfaces, optical forces, beam shifts upon reflection at a dielectric interface, and Wigner time delays in 1D scattering. This unifying framework illuminates the interplay between geometry and dynamics in wave scattering and can be applied to a broad range of physical systems.