Dynamic and geometric shifts in wave scattering.

Since Berry's pioneering 1984 work, the separation ofgeometricanddynamiccontributions in thephaseof 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 ofwave scatteringproblems, where observables (such as frequency, momentum, or position) experienceshifts in their expectation valuesbetween the input and output wave states. We describe this class of problems using a unitary scattering matrix and the associatedgeneralized Wigner-Smith operator(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 theeigenvaluesandeigenvectorsof 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.