New aspects of inverse scatterng theory on non-compact manifolds-from lattices to orbifolds

• Project/Area Number: 25287016
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: University of Tsukuba
• Principal Investigator: Isozaki Hiroshi 筑波大学, 数理物質系, 教授 (90111913)
• Co-Investigator(Kenkyū-buntansha): YAMAMOTO Masahiro 東京大学, 数理物質科学研究科, 教授 (50182647)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥12,350,000 (Direct Cost: ¥9,500,000、Indirect Cost: ¥2,850,000); Fiscal Year 2015: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2014: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2013: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
• Keywords: 逆問題 / シュレーディンガー作用素 / S行列 / ディリクレーノイマン写像 / 境界制御法 / 散乱理論 / リーマン多様体 / オービフォールド / スペクトル理論 / 格子

Outline of Final Research Achievements

On some non-compact manifolds with general metric, we have proven that the knowledge of S-matrix of one fixed end determines the whole Riemannian metric. The behavior of the metric at infinity is general enough to include all natural metrics. It also includes orbifolds appearing in number theory. We have proven that from the S-matrix of the Schroedinger operator on perturbed periodic lattices which include the case of graphen we can recover the compactly supported potentials and/or defects of the lattice. We have also proven that from the scattering operator of the Maxwell equation in an exterior domain with arbitrary anisotropic medium in a bounded part we can determine the 1st Bettii number of the boundary.

Research Products

• [Int'l Joint Research] Marseille 大学(フランス)
• [Int'l Joint Research] London 大学(英国)
• [Int'l Joint Research] Helsinki 大学(フィンランド)
• [Journal Article] Inverse scattering on multi-dimensional asymptotically hyperbolic orbifold (2015)
  • Author(s): H. Isozaki, Y. Kurylev and M. Lassas
  • Journal Title: Contemporary Mathematics Volume: 640 Pages: 71-85
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Inverse scattering at a fixed energy for discrete Schroedinger operators on the square lattice (2015)
  • Author(s): H.Isozaki and H. Morioka
  • Journal Title: Ann. l’Inst. Fourier Volume: ６５ Pages: 1153-1200
  • Peer Reviewed
• [Journal Article] Inverse problems for time dependent singular heat conductivities ; Multi-dimensional case (2015)
  • Author(s): P.Gaitan, H.Isozaki, O.Poisson, S.Siltanen and J.Tamminenn
  • Journal Title: Comm. in PDE Volume: 40 Pages: 837-877
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Inverse problems for time-dependent singular heat conductivities (2015)
  • Author(s): P.Gaitan, H.Isozaki, O.Poisson,S>Siltanen,J.Tamminen
  • Journal Title: Comm. P. D. E. Volume: 40 Pages: 837-877
  • Peer Reviewed / Open Access
• [Journal Article] Recent progress of inverse scattering theory on non-compcat manifolds (2014)
  • Author(s): H. Isozaki, Y. Kurylev, M. Lassas
  • Journal Title: Contemp. Math Volume: 615 Pages: 143-163
  • DOI: 10.1090/conm/615/12290
  • Peer Reviewed / Open Access
• [Journal Article] A Rellch type theoerm for discrete Schroedinger operators (2014)
  • Author(s): H. Isozkai, H. Morioka
  • Journal Title: Inverse Prbl. Imaging Volume: 8 Issue: 2 Pages: 475-489
  • DOI: 10.3934/ipi.2014.8.475
  • Peer Reviewed / Open Access
• [Journal Article] Inverse problems for time-dependent singular heat conductivities - One-diemensional case (2013)
  • Author(s): P.Gaitan, H.Isozaki, O.Poisson, S.Siltanen and J.Tamminen
  • Journal Title: SIAM J. Math. Anal. Volume: 45 Issue: 3 Pages: 1675-1690
  • DOI: 10.1137/120886510
  • Peer Reviewed
• [Presentation] Inverse scattering for Schroedinger operators on perturbed periodic lattices (2016)
  • Author(s): 磯崎 洋
  • Organizer: RIMS シンポジウム 偏微分方程式の逆問題とその応用の新展開
  • Place of Presentation: 京都大学数理解析研究所 (京都府京都市）
  • Year and Date: 2016-01-28
  • Invited
• [Presentation] Asymptotic properties of solutions to the elastic equation in a half-space (2015)
  • Author(s): 磯崎 洋
  • Organizer: CIRM Workshop, Control of PDE’s and Application
  • Place of Presentation: CIRM （France, Marseille）
  • Year and Date: 2015-11-12
  • Int'l Joint Research / Invited
• [Presentation] Inverse scattering on non-compact manifolds with general metriｃ (2015)
  • Author(s): 磯崎 洋
  • Organizer: Modern theory of wave equations
  • Place of Presentation: SchroedingerInstitute (Austria, Wien)
  • Year and Date: 2015-08-28
  • Int'l Joint Research / Invited
• [Presentation] Inverse scattering on non-compact manifolds with general metric (2015)
  • Author(s): 磯崎 洋
  • Organizer: Spectral and analytic inverse problems
  • Place of Presentation: Henri Poincare Institute (France, Paris)
  • Year and Date: 2015-05-05
  • Int'l Joint Research / Invited
• [Presentation] Inverse scattering on non-compact manifolds with general metric (2015)
  • Author(s): Hiroshi Isozaki
  • Organizer: Math. Physics Conference
  • Place of Presentation: University of Nantes (France)
  • Year and Date: 2015-02-03
  • Invited
• [Presentation] Inverse scattering on non-compact manifolds with general metric (2014)
  • Author(s): Hiroshi Isozaki
  • Organizer: Probleme inverses et domaine associe
  • Place of Presentation: Aix-Marseille University (France)
  • Year and Date: 2014-11-28
  • Invited
• [Presentation] Spectral properties for Laplacians on non-compact manifolds with general metric (2014)
  • Author(s): Hiroshi Isozaki
  • Organizer: Inverse problems and related topics
  • Place of Presentation: Euler Institute (Russia)
  • Year and Date: 2014-08-19
  • Invited
• [Presentation] Spectral theory and inverse problems for Laplacians on perturtbed lattices (2014)
  • Author(s): Hiroshi Isozaki
  • Organizer: AIMS Conference
  • Place of Presentation: Madrid (Spain)
  • Year and Date: 2014-07-09
  • Invited
• [Presentation] Spectral theory and inverse problems for Laplacians on perturtbed lattices (2014)
  • Author(s): Hiroshi Isozaki
  • Organizer: REcent progress in mathematical and numerical analysis of inverse problems
  • Place of Presentation: Luminy, Marseille (France)
  • Year and Date: 2014-05-19
  • Invited
• [Presentation] Inverse scattering on pertrurbed lattices
  • Author(s): H. Isozaki
  • Organizer: Workshop "Inverse Problems and Applications"
  • Place of Presentation: Mittag-Leffler Institute (Sweden)
  • Invited
• [Presentation] Inverse scattering on asymptotically hyperbolic orbifolds
  • Author(s): H. Isozaki
  • Organizer: Spectral Theory and Partial Differential Equations
  • Place of Presentation: UCLA (USA)
  • Invited
• [Presentation] Inverse problems for time-dependent heat conductivities in multi-dimensions
  • Author(s): H. Isozaki
  • Organizer: Applied Inverse Problems Conference at KAIST
  • Place of Presentation: Daejeon (Korea)
  • Invited
• [Presentation] オービフォールド・カスプ・逆散乱
  • Author(s): 磯崎 洋
  • Organizer: 実函数論・函数解析合同シンポジウム
  • Place of Presentation: 青山学院大学
  • Invited
• [Presentation] Inverse spectral theory and the Minkowski problem for the surface of revolution
  • Author(s): 磯崎 洋
  • Organizer: 広島微分方程式研究会
  • Place of Presentation: 広島大学
  • Invited
• [Book] Introduction to spectral theory and inverse problems on asymptotically hyperbolic manifolds (2014)
  • Author(s): H. Isozaki, Y.Kurylev
  • Total Pages: 251
  • Publisher: Mathematical Society of Japan
• [Funded Workshop] 偏微分方程式姫路研究集会 (2016)
  • Place of Presentation: イーグレ姫路 (兵庫県姫路市）
  • Year and Date: 2016-03-02