Inverse problems for Schroedinger equations and wave equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03617
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Okayama University of Science (2020-2023); Niigata University (2019)
• Principal Investigator: Watanabe Michiyuki 岡山理科大学, 理学部, 教授 (90374181)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 逆問題 / 散乱理論

Outline of Final Research Achievements

Important progress was made in the mathematical study of inverse scattering problems in quantum mechanics and inverse problems for the identification of nonlinearities. In the final year, new methods were developed to apply the Hartree and high-energy Born approximation methods to the two-dimensional case in space, and throughout the entire research period, new analytical methods were established for the nonlinear wave equation and the nonlinear Schroedinger equation. These results are expected not only to contribute to the elucidation of piezoelectric application techniques and atomic structure, but also to have a significant impact on the future development of mathematical research.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

本研究の成果は、圧電体の利用技術の進展や原子構造の解明に寄与する可能性があります。例えば、ライターやコンロ、ソナーやスピーカーなどの技術における圧電体の応用に役立つだけでなく、原子構造の解明においても新たな知見を提供することが期待されます。特に、ハートリー・ホック近似法と高エネルギーボルン近似法を組み合わせた新しい考え方は、従来の手法では困難だった問題の解決に寄与し、量子力学における逆散乱問題の理解を深める重要なステップとなりました。今後は、より多様な数理モデルに対してこれらの手法を検証し、応用範囲を広げる予定です。