Spectral theory of Neumann- Poincare operators and its Generalization

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K13805
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Shinshu University
• Principal Investigator: Miyanishi Yoshihisa 信州大学, 学術研究院理学系, 准教授 (20740236)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: ノイマン・ポアンカレ作用素 / 積分作用素 / スペクトル

Outline of Final Research Achievements

We obtained the spectral structure of integral operators (So-called Neumann-Poincare operators) for several domains. Such structure deeply depends on the boundary geometry and we proved the precise asymptotics even for boundaries which are smoother than C^2. It is emphasized that the techniques employed in our papers can be applied for many general singular integral operators. Then the applications are considered in physics and pure mathematics. In physics, plasmons of electro-static phenomena are controlled by the spectral structure. In pure mathematics, we also prove the relation between algebraic structure (Partition of an unit interval) and the spectrum.

Free Research Field

大域解析学

Academic Significance and Societal Importance of the Research Achievements

本研究はとくに、スペクトル解析の進んでいる擬微分作用素と呼ばれる作用素を特異積分作用素の近似として用いることで、（自己共役でもない）計算の困難な特異積分作用素のスペクトル構造を得る方法を提案している。また、具体的な現象に対応したノイマン・ポアンカレ作用素と呼ばれる作用素のスペクトルによって、３次元(実空間における)の電磁気現象など物理現象の解明にも繋っている。また、積分作用素のスペクトル構造を、代数や幾何にまで応用できることも示している。