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Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

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Published on: May 27, 2020

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator.

Stefan Klus¹, Feliks Nüske², Boumediene Hamzi³

¹Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany.

Entropy (Basel, Switzerland)

|December 8, 2020

Summary

This study introduces a kernel-based method to approximate differential operators and estimate eigenfunctions from data. This approach aids in dimensionality reduction for molecular dynamics and quantum chemistry, linking quantum mechanics to stochastic processes.

Keywords:

Koopman generator Schrödinger operator reproducing kernel Hilbert space

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Area of Science:

Computational physics
Quantum chemistry
Data-driven modeling

Background:

Dimensionality and model reduction techniques often require estimating eigenfunctions of dynamical operators.
Key examples include the Koopman operator and the Schrödinger operator, crucial in various scientific domains.

Purpose of the Study:

To propose a novel kernel-based method for approximating differential operators.
To demonstrate the estimation of eigenfunctions via auxiliary matrix eigenvalue problems.
To bridge quantum mechanics and stochastic differential equations.

Main Methods:

Kernel-based approximation of differential operators in reproducing kernel Hilbert spaces.
Solving auxiliary matrix eigenvalue problems for eigenfunction estimation.
Transforming the Schrödinger operator into a Kolmogorov backward operator.

Main Results:

Successfully applied algorithms to molecular dynamics and quantum chemistry.
Established a transformation between the Schrödinger operator and the Kolmogorov backward operator.
Enabled the application of stochastic differential equation methods to quantum systems.

Conclusions:

The proposed kernel-based method offers an effective way to approximate differential operators and their eigenfunctions.
The established link between quantum mechanics and stochastic processes opens new avenues for analysis.
This work facilitates advanced data-driven modeling in computational physics and chemistry.