Avoiding spectral pollution for transfer operators using residuals

TL;DR

This paper introduces algorithms to accurately compute the spectral properties of transfer operators in dynamical systems, avoiding spectral pollution and applicable to various models including molecular dynamics.

Contribution

The paper develops new algorithms that prevent spectral pollution in transfer operator spectral analysis, extending to Hardy-Hilbert spaces and addressing a key challenge in the field.

Findings

Algorithms successfully avoid spectral pollution in transfer operator analysis.

Case studies demonstrate high accuracy and flexibility of the methods.

Spectral features can exist outside the chosen function space, revealing subtle analytic issues.

Abstract

Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.