Semi-classical analysis of the Schroedinger equations

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04971
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Ritsumeikan University
• Principal Investigator: FUJIIE Setsuro 立命館大学, 理工学部, 教授 (00238536)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 準古典解析 / 超局所解析 / シュレディンガー作用素 / 量子共鳴 / エネルギー交差

Outline of Final Research Achievements

The main results of the research during this period are the following three. (i) We established a propagation of singularity theorem of the quantum system at a hyperbolic fixed point of the corresponding classical mechanics. We proved that if the semiclassical wave front set is empty on the incoming stable manifold associated to the fixed point, then it is also empty on the outgoing stable manifold. (ii) We prove a theorem that the spectral shift function of the Schroedinger operator with matrix-valued potential has a complete asymptotic expansion if its symbol has a scalar escape function. (iii) We clarified the semiclassical asymptotic distribution of resonances of the Schroedinger operator with matrix-valued potential for a model of one-dimensional 2 by 2 system with an energy level crossing.

Free Research Field

偏微分方程式