Generalized eigenfunction expansion for the Lorenz system's Koopman operator

Determine a generalized eigenfunction expansion for the unitary Koopman operator associated with the Lorenz (63) system on its SRB-measure-supported attractor, specifying the generalized eigenfunctions and corresponding spectral measures in a rigged Hilbert space that yield a mode decomposition for observables despite the continuous spectrum on the unit circle except at λ = 1.

Background

Within the paper’s examples, the authors highlight that the Lorenz system has a continuous spectrum on the unit circle (excluding the trivial eigenvalue at 1), which complicates modal analysis using standard eigenfunction-based approaches. They note that while Rigged DMD can compute wave-packet approximations and construct a rigged Hilbert space via time-delay embedding, an explicit generalized eigenfunction expansion for this system is not available.

Establishing such an expansion would provide a formal decomposition of observables into generalized eigenfunctions and associated spectral measures for the Lorenz system, analogous to the expansions the authors develop and apply to other classes of systems in the paper. This would clarify the structure of coherent features in the Lorenz dynamics under the Koopman framework and validate the rigged Hilbert space approach for a canonical chaotic system.