Semiclassical analysis of sprectral and scattering problems on energy-level crossings
| Project/Area Number |
18K03349
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Ritsumeikan University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | エネルギー交差 / 準古典解析 / 量子共鳴（レゾナンス） / 遷移確率 / 完全WKB解析 / 準古典超局所解析 / レゾナンス / 超局所解析 / WKB解析 |
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| Outline of Final Research Achievements |
The keyword of this research project is "energy-level crossing". Concerning semiclassical distributions of quantum resonances, our series of works (Fujiie-Martinez-Watanabe) showed the existence and distribution of quantum resonances whose imaginary part is a fractional order with respect to the semiclassical parameter. It can be understood that these resonances are created by a weakly trapped trajectories, which cannot be realized in the case of a single-valued Schrodinger operator.
We also studied a problem of an adiabatic transition probability for avoided crossings and, in particular, a problem to which can not be applied an exact WKB method. The problem in a linear crossing model was solved by Watanabe-Zerzeri by means of a reduction to a microlocal normal form and that in a tangential crossing model has been investigated by Higuch-Watanabe thanks to the recent approach introduced by Assal-Fujiie-Higuchi.
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| Academic Significance and Societal Importance of the Research Achievements |
量子共鳴や遷移確率は、スペクトル・散乱理論の研究対象であり、特に準古典解析として、古典軌道といった古典力学特有の性質により量子力学の現象を記述する典型的なトピックである。
本研究の学術的意義は、連立系を扱うことにより、単独の場合には現れない古典軌道をもつシュレディンガー作用素を考察することが可能になり、実際に単独の場合では得られない量子共鳴の存在を明らかにした点である。特に、既存の完全WKB解析や準古典超局所解析の手法では解析できない場合に対する新たな手法を提案したことの意義も大きく、我々の手法を拡張した最近の結果は、もう一方の研究課題であるLandau-Zenerの公式の一般化にも適用された。
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Report
(6 results)
Research Products
(43 results)
