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

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.