Bibliographic Information: Glazatov, V. A., & Sakbaev, V. Zh. (2024). On the extension of singular linear infinite-dimensional Hamiltonian flows. Lobachevskii Journal of Mathematics. arXiv:2410.06749v1 [math-ph]
Research Objective: This paper investigates Hamiltonian flows in infinite-dimensional Hilbert spaces, focusing on systems exhibiting singular behavior like infinite kinetic energy in finite time. The authors aim to develop a method for extending the analysis of such systems beyond the point of singularity.
Methodology: The researchers utilize the concept of symplectic measures, constructing a new class of measures invariant under specific Hamiltonian flows. They extend the phase space from a Hilbert space to a locally convex space of numerical sequences, allowing for the continuation of phase trajectories beyond their initial domain. The Koopman representation is then employed to analyze the extended flow.
Key Findings: The paper demonstrates that the introduced symplectic measures remain invariant under the extended Hamiltonian flow in the expanded phase space. This allows for the definition of a Koopman unitary representation of the flow, even for systems exhibiting singular behavior.
Main Conclusions: The research provides a new framework for analyzing Hamiltonian systems with singular flows. By extending the phase space and employing invariant measures, the authors offer a method to continue the analysis beyond the singularity point, opening possibilities for a deeper understanding of such systems.
Significance: This work contributes significantly to the field of infinite-dimensional Hamiltonian systems, particularly those exhibiting singular behavior. The introduced methodology can be applied to various physical problems, including those related to gradient explosions in evolutionary PDEs.
Limitations and Future Research: The paper focuses on a specific class of Hamiltonian systems. Further research could explore the applicability of this method to a broader range of Hamiltonian systems and investigate the properties of the Koopman representation in more detail.
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by Vladimir Gla... at arxiv.org 10-10-2024
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