Koopman analysis of quantum systems

TL;DR

This paper explores the application of Koopman operator theory and data-driven methods to analyze quantum systems, establishing connections with stochastic control and Schrödinger equations to open new research avenues.

Contribution

It introduces a novel approach linking Koopman analysis with quantum mechanics, demonstrating how data-driven techniques can solve Schrödinger equations.

Findings

Koopman methods can analyze quantum systems effectively.

Data-driven stochastic control approaches solve Schrödinger equations.

New connections between dynamical systems and quantum physics are established.

Abstract

Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schr\"odinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schr\"odinger equation. Our findings open up a new avenue towards solving Schr\"odinger's equation using recently developed tools from data science.