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Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

by Stefan Klus 1,*,†, Feliks Nüske 2,† and Boumediene Hamzi 3
1
Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany
2
Department of Mathematics, Paderborn University, 33098 Paderborn, Germany
3
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Entropy 2020, 22(7), 722; https://doi.org/10.3390/e22070722
Received: 27 May 2020 / Revised: 25 June 2020 / Accepted: 26 June 2020 / Published: 30 June 2020
Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems. View Full-Text
Keywords: Koopman generator; Schrödinger operator; reproducing kernel Hilbert space Koopman generator; Schrödinger operator; reproducing kernel Hilbert space
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Klus, S.; Nüske, F.; Hamzi, B. Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator. Entropy 2020, 22, 722.

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