Kernel-based approximation of the Koopman generator and Schrödinger operator

Stefan Klus*, Feliks Nüske, Boumediene Hamzi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)
32 Downloads (Pure)


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.

Original languageEnglish
Article number722
Issue number7
Publication statusPublished - Jul 2020


  • Koopman generator
  • Reproducing kernel hilbert space
  • Schrödinger operator

ASJC Scopus subject areas

  • Information Systems
  • Mathematical Physics
  • Physics and Astronomy (miscellaneous)
  • Electrical and Electronic Engineering


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