Singular nature in nearly integrable Hamiltonian systems and breakdown of classical-quantum correspondence

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05583
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical physics/Fundamental condensed matter physics
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: Shudo Akira 東京都立大学, 理学研究科, 教授 (60206258)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 近可積分ハミルトン系 / 量子カオス / トンネル効果 / 複素古典力学 / 遅い緩和 / Perron-Frobenius演算子 / 脱出率 / ガラス

Outline of Final Research Achievements

1) The tunneling effect in non-integrable systems was analysed on the basis of the complex stable manifold mechanism. It is found that the complex orbits that contribute most to the tunneling effect are along complex stable manifolds that asymptotically approach the unstable periodic orbit closest to the regular domain. It is found that anomalous tunneling effects also appear in ultra-near integrable systems that have no structures inherent in nonintegrability of the classical system.
2) The slow relaxation in the Hamiltonian system without regular structures in classical phase space was considered. The relaxation problem was discussed by studying the second eigenfunction of the Perron-Frobenius operator of a discrete map, the escape rate from a closed domain and the dynamical origin of the slow relaxation in glass, are analysed, especially focusing on the correction from the time going infinity limit.

Free Research Field

非線形物理学

Academic Significance and Societal Importance of the Research Achievements

近可積分ハミルトン系は，ニュートン力学に支配される物理現象のほぼすべてを含む力学系のカテゴリーである．本研究は，世の中に最もありふれた系の古典力学および量子力学に対する基本的性質を明らかにすることを目的としたものである．前者に関しては，世の中の多くの現象がなぜ直ちに熱平衡に向かうことがないか？また，後者については，非可積分な系の量子力学固有のトンネル効果が可積分な系のそれと本質的にどこが違うのか？という点について考察したものであり，自然現象に対する理解を最も基礎的な原理から考察した研究として位置づけられる．