Related papers: Numerical approximation of the Koopman-von Neumann…
It has long been known that there exists a coordinate transformation which exactly maps the quantum free particle to the quantum harmonic oscillator. Here we extend this result by reformulating it as a unitary operation followed by a time…
Koopman theory studies dynamical systems in terms of operator theoretic properties of the Perron-Frobenius operator $\mathcal{P}$ and Koopman operator $\mathcal{U}$ respectively. In this paper, we derive the rates of convergence of…
The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical…
The Koopman operator approach provides a powerful linear description of nonlinear dynamical systems in terms of the evolution of observables. While the operator is typically infinite-dimensional, it is crucial to develop finite-dimensional…
Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum coupling. The proposed model not only…
The Koopman operator framework holds promise for spectral analysis of nonlinear dynamical systems based on linear operators. Eigenvalues and eigenfunctions of the Koopman operator, so-called Koopman eigenvalues and Koopman eigenfunctions,…
We develop a Koopman operator framework for studying the {computational properties} of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ``Koopman…
The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct…
Transfer operators offer linear representations and global, physically meaningful features of nonlinear dynamical systems. Discovering transfer operators, such as the Koopman operator, require careful crafted dictionaries of observables,…
We provide an overview of the Koopman operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the…
Koopman analysis of a general dynamics system provides a linear Koopman operator and an embedded eigenfunction space, enabling the application of standard techniques from linear analysis. However, in practice, deriving exact operators and…
Koopman operator theory offers a rigorous treatment of dynamics and has been emerging as an alternative modeling and learning-based control method across various robotics sub-domains. Due to its ability to represent nonlinear dynamics as a…
In a previous article [H. Bergeron, J. Math. Phys. 42, 3983 (2001)], we presented a method to obtain a continuous transition from classical to quantum mechanics starting from the usual phase space formulation of classical mechanics. This…
Complex eigenspectra of transfer and Koopman operators describe rotational motion in dynamical systems. A particularly relevant situation in applications is when the rotation speed depends on the state-space position of the dynamics. We…
The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches…
In this paper we continue the study, started in [1], of the operatorial formulation of classical mechanics given by Koopman and von Neumann (KvN) in the Thirties. In particular we show that the introduction of the KvN Hilbert space of…
Representing nonlinear dynamical systems using the Koopman Operator and its spectrum has distinct advantages in terms of linear interpretability of the model as well as in analysis and control synthesis through the use of well-studied…
Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used…
Quantum computation offers potential exponential speedups for simulating certain physical systems, but its application to nonlinear dynamics is inherently constrained by the requirement of unitary evolution. We propose the quantum Koopman…
This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum…