Scattering theory for periodic Schroedinger Operators

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400165
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kyoto Institute of Technology
• Principal Investigator: Mine Takuya 京都工芸繊維大学, 基盤科学系, 准教授 (90378597)
• Research Collaborator: NOMURA Yuji ; KAMINAGA Masahiro
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 数理物理学 / 量子力学 / 大域解析学 / アハラノフ・ボーム効果 / シュレディンガー方程式 / 分散型評価

Outline of Final Research Achievements

(1) We give an explicit formula for the scattering amplitude for the Schroedinger operator with two-point delta-like magnetic fields satisfying the magnetic quantization condition, and give a numerical calculation for the amplitude. The obtained results are consistent with the asymptotic formula by Ito and Tamura. Moreover, we give an explicit formula for the spectral shift function for that model.
(2) We give a dispersive estimate for the Kronig-Penney model, which is an explicitly solvable model for the one-dimensional periodic quantum system. The estimate consists of two terms; one has the usual decay t to the power -1/2, and another has the slower decay t to the power -1/3. We also give a decay estimate for the coefficient of the latter term, with respect to the band number of the model.

Free Research Field

数物系科学