Semi-classical analysis of Schroedinger equations

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540196
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Ritsumeikan University
• Principal Investigator: FUJIIE Setsuro 立命館大学, 理工学部, 教授 (00238536)
• Co-Investigator(Renkei-kenkyūsha): IWASAKI Chisato 兵庫県立大学, 大学院・物質理学研究科, 教授 (30028261) ; UMEDA Tomio 兵庫県立大学, 大学院・物質理学研究科, 教授 (20160319) ; YAMADA Osanobu 立命館大学, 理工学部, 教授 (70066744) ; WATANABE Takuya 立命館大学, 理工学部, 准教授 (80458009)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 準古典解析 / シュレディンガー作用素 / 超局所解析 / WKB解析 / レゾナンス / スペクトルシフト関数

Outline of Final Research Achievements

First, I wrote down the quantization condition of resonances and clarified the asymptotic distribution of these resonances for the Schroedinger operator near an energy at which the corresponding classical mechanics has a non-degenerate hyperbolic fixed point and associated homoclinic or heteroclinic orbits as trapped set.
Second, I proved that the spectral shift function for the Stark hamiltonians has a complete asymptotic expansion at non-trapping energies.
Finally, I wrote down the quantization condition of resonances and clarified the asymptotic distribution of these resonances near a crossing level of two potentials of a system of Schroedinger operators.

Free Research Field

偏微分方程式論