Across mathematics, physics, biology and geoscience, along with nonlinear quantum technologies, multi-hazards and systems intelligence, we work with complex systems exhibiting nonlinear, non-normal, multiscale, transient behaviour, sensitive to perturbations and shaped by emergence. The world does not revolve around stability. It revolves around complexity.

However, most classical systems analysis, model design and computational protocols are built around stability concepts, eigenvalues, eigenfunctions, invariant modes, and diagonalizable dynamics. While adequate for dynamic “business as usual”, they are not sufficiently compelling in dealing with the challenging complexity of a changing world.
For example, typical AI approaches implicitly search for compact representations, stable latent spaces, or dominant modes. Yet real-world systems require a more adequate encoding of instability, emergence, transition, amplification, and extremes.

These call for a foundational mathematical paradigm shift.
One of my contributions to this effort formulates “Resolvent-Generated Generalized Spectral Operators for Nonlinear Dynamical Systems via Koopman Semigroups” (Perdigão, 2026).

In theoretical mathematics, my work contributes to the interface between semigroup theory, resolvent analysis, generalized spectral theory, functional calculus, Koopman operator theory, and non-normal operator theory. It proposes a way to build spectral representations from the resolvent itself, giving a rigorous path toward nonlinear spectral analysis when eigenfunctions alone are not enough.

In operator theory and dynamical systems, it helps clarify how spectral information can be organized in settings where the generator and Koopman semigroup may not admit a simple eigenbasis. This is especially relevant for nonlinear flows, infinite-dimensional observables, continuous spectra, and weak operator limits.

With physics in mind, it offers a language for systems where transient growth, resonances, instabilities, and non-normal amplification may matter as much as asymptotic modes. This is relevant to e.g. fluid mechanics, plasma dynamics, nonlinear wave phenomena, far-from-equilibrium processes, and multi-hazard dynamics.

With quantum applications in mind, it empowers the analysis of evolution where spectral structure is not exhausted by stationary states. Open quantum systems, resonances, decoherence, scattering, and non-Hermitian or effective descriptions often require tools that go beyond simple normal spectral decompositions.

With AI, scientific machine learning, and physical systems intelligence in mind, it improves the mathematical foundations for Koopman-based learning, operator learning, reduced-order modeling, and interpretable dynamics. It is especially relevant when finite-dimensional spectral approximations may be affected by non-normality, continuous spectra, or spectral pollution.

By Rui A. P. Perdigão, May 19th, 2026

Reference:

Perdigão, R.A.P. (2026): Resolvent-Generated Generalized Spectral Operators for Nonlinear Dynamical Systems via Koopman Semigroups. Mathematics 2026, 14(7), 1145; https://doi.org/10.3390/math14071145

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