New develpment of spectral and inverse scattering theory-Non linear problems and continuum limit

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03667
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Ritsumeikan University
• Principal Investigator: Isozaki Hiroshi 立命館大学, 総合科学技術研究機構, プロジェクト研究員 (90111913)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 逆問題 / S行列 / ディリクレ・ノイマン写像 / シュレーディンガー作用素 / 離散グラフ

Outline of Final Research Achievements

To know the characteristics of our ambient space or physical system by observing the wave propagation is the most fundamental problem in our recognition of the world. In this research, we studied the mathematical properties of various spectral quantities related to the waves on these manifolds and the associated inverse problems to recover the system in question. Our scope ranges over not only continuous manifolds but also on discrete manifolds, i.e. discrete graphs. We solved the inverse scattering problem on non-compact Riemannnian manifolds with general metric, the stationary scattering theory on the elastic equation in the half-space, the inverse problem for Laplacians on discrete graphs, as well as the inverse scattering problem on locally perturbed periodic lattices. We also obtained the asymptotic expansion of solutions to the stationary elastic wave equation in the 3-dimensional half-space and derived the Rayleigh wave propagating only along the surface.

Free Research Field

スペクトル理論と逆問題

Academic Significance and Societal Importance of the Research Achievements

今日MRI等の画像診断は医療に不可欠なものとなっている.工学的問題において建築物の構造診断の際に音響診断のみならずサーモグラフィー等による遠隔からの非破壊的方法も極めて重要かつ有効である.このような観測データの解析から正しい結果が判定できるかどうかはその背後に確固たる理論的基礎がある場合のみであり、そのための理論的基礎、例えば解の一意性の問題、物理的パラメータの再構成のアルゴリズム、その安定性等を構築するのがこの研究の目的である. それは既知の数学の応用にとどまらず新しい数学的問題と手法の発見、既存の方法の深化等の理論的発展もうながすとともに数値計算にも重要なインパクトを与えるものである.