2019 Fiscal Year Final Research Report

New models of inverse spectral and scattering theory - form discrete to condinuoud

Research Project

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Project/Area Number	16H03944
Research Category	Grant-in-Aid for Scientific Research (B)
Allocation Type	Single-year Grants
Section	一般
Research Field	Mathematical analysis
Research Institution	Ritsumeikan University
Principal Investigator	Isozaki Hiroshi 立命館大学, 理工学部, 授業担当講師 (90111913)
Co-Investigator(Kenkyū-buntansha)	岩塚明京都工芸繊維大学, 基盤科学系, 教授 (40184890)
Project Period (FY)	2016-04-01 – 2020-03-31
Keywords	逆問題 / S行列 / ディリクレーノイマン写像 / リーマン計量 / 格子 / シュレーディンガー作用素
Outline of Final Research Achievements	(1) In a most general class of non-compact Riemannian manifolds with ends equipped with prescribed metrics at infinity, I have solved the inverse scattering problem: From one component of the S-matrix associated with an arbitrary end, one can reconstruct the Riemannian metric and the topology of whole manifold. One can allow asymptotically hyperbolic and polynomially growing or decaying ends (hence one can allow cusps), and also the conic singularities appearing in the orbifolds. (2) On locally perturbed periodic lattices, I solved the inverse scattering problem. Given a S-matrix, one can reconstruct the perturbation of the lattice. It contains the physically imporant example of graphene.I have also solved the inverse scattering problem for quantum graph.(3) For the boundary value problem of the elastic wave equation in a half-space, I have derived the asymptptotic expansion of the reduced wave equation at infinity. It contains the Reyleigh waves propagating along the surface.
Free Research Field	数学，解析学
Academic Significance and Societal Importance of the Research Achievements	逆問題の目標は直接の観測が困難な対象を間接的情報から推測，同定，再構成することにあり，その応用は原子・分子等のミクロな物理の世界から，工学における非破壊検査，X線トモグラフィー等の医療, さらに資源探査等にまで広く及んでいる．この逆問題の理論的背景を解明することは，応用上の成果に理論的支柱を与えると共に，新しい応用も示唆する．リーマン多様体上の逆問題は数学の世界での大きな問題であるが，さらに格子上の逆問題を考えることによって，数学の中の純理論的考察と並行したことが固体物理の世界にも適用できることを示した．本研究は離散と連続に共通した逆問題研究の方法があることを示したことでも意義深い．