Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910878285168640 |
|---|---|
| author | Stengl, Marian Gelß, Patrick Klus, Stefan Pokutta, Sebastian |
| author_facet | Stengl, Marian Gelß, Patrick Klus, Stefan Pokutta, Sebastian |
| contents | The Koopman--von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman--von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set's closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman--von Neumann framework to transport equations is highlighted. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13504 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains Stengl, Marian Gelß, Patrick Klus, Stefan Pokutta, Sebastian Analysis of PDEs Mathematical Physics Dynamical Systems Functional Analysis 35A05, 35F10, 37C30, 46E35, 47D06 The Koopman--von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman--von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set's closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman--von Neumann framework to transport equations is highlighted. |
| title | Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains |
| topic | Analysis of PDEs Mathematical Physics Dynamical Systems Functional Analysis 35A05, 35F10, 37C30, 46E35, 47D06 |
| url | https://arxiv.org/abs/2306.13504 |