Spectral asymptotic analysis for quantum phenomena related to magnetic fields

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05230
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Okayama University
• Principal Investigator: Tamura Hideo 岡山大学, 自然科学研究科, 特命教授 (30022734)
• Co-Investigator(Kenkyū-buntansha): 筧 知之 筑波大学, 数理物質系, 教授 (70231248) ; 岩塚 明 京都工芸繊維大学, 基盤科学系, 教授 (40184890)
• Research Collaborator: Yajima Kenji ; Nakamura Shu ; Mine Takuya
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: スペクトル理論 / シュレディンガー作用素 / 磁場散乱 / レゾナンス / アハラノフ・ボーム効果 / ２次元エフィモフ効果 / 3体フェルミ系

Outline of Final Research Achievements

We study the Aharonov--Bohm effect (the AB effect) in quantum resonances for scattering by three solenoids in two dimensions under the situation that the centers of the solenoids are placed almost in line. The resonances are shown to be generated near the real axis by the trapping trajectories when the centers are largely separated from one another. We analyze how the AB effect is reflected in the location of these resonances. The results are described by use of the backward scattering amplitude by one solenoid and depend on the ratio of the distances between the solenoids as well as on the magnetic fluxes. The method makes a full use of the information from the scattering system by one solenoid, which is known to be exactly solvable in quantum mechanics. We also discuss the case of four solenoids.

Free Research Field

数学解析

Academic Significance and Societal Importance of the Research Achievements

量子力学の研究対象であるミクロな世界においては, 古典力学の直感とは相容れない量子効果とよばれる現象が生じる。本研究課題では量子力学の根幹に関る現象のひとつであるアハラノフ・ボーム効果 (AB 効果) をスペクトル漸近解析を駆使して研究した。AB 効果とは, 量子力学に従う粒子が磁場の中を運動するとき，磁場のみならず, それを生成するベクトル ポテンシャル自身も運動に関与する現象を言う。ほぼ直線上に並んだ３個のソレドノイド磁場による散乱系において, 左右両端の磁場間の捕捉現象から生じるレゾナンスに第三の磁場の磁束と配置が関係することを示し, AB 効果の実在を数学的に立証した。