Avoiding spectral pollution for transfer operators using residuals

• URL: http://arxiv.org/abs/2507.16915v1
• Date: Tue, 22 Jul 2025 18:01:05 GMT
• Title: Avoiding spectral pollution for transfer operators using residuals
• Authors: April Herwig, Matthew J. Colbrook, Oliver Junge, Péter Koltai, Julia Slipantschuk,
• Abstract summary: We present algorithms for computing spectral properties of transfer operators without spectral pollution.<n>Case studies range from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding.<n>Our methods offer robust tools for spectral estimation across a broad range of applications.
• Score: 0.6116681488656472
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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