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

2015 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

散乱理論