Prove the near-constancy of the resonant-subspace projector

Prove that the projector onto the resonant subspace of the round-trip matrix remains approximately constant under variations of photon energy, wavevector, geometric dimensions, permittivities, and other parameters within the local parameter region used by the resonant subspace approximation.

Background

The resonant subspace approximation replaces the parameter-dependent resonant eigenvectors of the round-trip matrix with a fixed basis evaluated at an anchor point. This replacement is justified by the assumption that the projector onto the resonant subspace varies only weakly across a local region of parameter space, even when individual resonant eigenvectors rotate rapidly near avoided crossings.

The paper explains that weak coupling between resonant modes and the non-resonant background is expected to make this projector nearly constant, thereby allowing the restricted round-trip matrix to be smoothly extrapolated. However, the paper explicitly acknowledges that this key assertion has not been proved. Establishing conditions under which the projector is approximately constant, and quantifying the resulting approximation error, would provide a rigorous foundation for the method.