Inverse scattering theory for Schroedinger operators by probabilistic method

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13447
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Ritsumeikan University (2016-2017); University of Tsukuba (2015)
• Principal Investigator: Isozaki Hiroshi 立命館大学, 理工学部, 授業担当講師 (90111913)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 逆散乱理論 / シュレーディンガー作用素 / S行列 / 確率分布

Outline of Final Research Achievements

The aim of this research is to derive probabilistic informations of potentials for discrete Schroedinger operators defined on lattices from the knowledge of the S-matrix, the fundamental physical quantity describing the scattering phenomena. The mesh size of the lattice is used as a parameter of approximation, and the ultimate goal is to obtain probabilistic distribution of of the potential randomly dsitributed on lattices. We have obtained estimates of the resolvent of free discrete Schroedinger operator uniform with respect to the mesh size. We are interested in the convergence of the resolvent of the discrete model to that of the continuous model, which proposes a new challenging problem for Schroedinger operators. We have found a new approximation scheme which possibly provides us with the solution. We also have another idea of approximation.

Free Research Field

数理物理学