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Interference and Surface Wave in Quantum Scattering

Research Project

Project/Area Number 08640248
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionSETSUNAN UNIVERSITY (1997)
Fukui National College of Technology (1996)

Principal Investigator

SHIMADA Shin-ich  Setsunan University, Depertment of Mathematics, Associate Professor, 工学部, 助教授 (40196481)

Co-Investigator(Kenkyū-buntansha) TSUBOKAWA Takehiro  Fukui National College of Technology, General Education, Associate Professor, 一般科目教室, 助教授 (70236941)
Project Period (FY) 1996 – 1997
Project Status Completed (Fiscal Year 1997)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1997: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1996: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsSchrodinger Operator / Scattering / Wave Operator / Wave Function / Interference / Scattering Amplitude / Schrodinger作用素 / 散乱 / 自己共役拡張
Research Abstract

Inspired by the Tonomura's famous experiment showing that the electron has the duality of wave and particle, we consider the Schrodinger operators with a singular potential supported only on a straight line L (x-axis) in 3-dimensional Euclidean space. Let A be a Laplacian with domain consisting of smooth functions with compact support off the line L.As a selfadjoint realization of A,we construct a selfadjoint operator H which is, in turn, a rank one perturbation of the free Hamiltonian H_O. H has at most one negative eigenvalue and no singularly continuos spectrum. The absolutely continuos spectrum agrees with non-negative real axis. We prove that the wave operators for the pair (H,H_O) exist and are complete. The scattering amplitude and wave functions can be calculated explicitly so that they are represented in 3-dimensinal graph. The interference appears in our model. Therefore H is seen to be a simple model describing the above mentioned experiment.
Our method can be applied to all the selfadjoint extensions of A.We study their spectral properties in detail.

Report

(3 results)
  • 1997 Annual Research Report   Final Research Report Summary
  • 1996 Annual Research Report
  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] 島田伸一: "A solvable model for line interaction" 京都大学数理解析研究所講究録. 994. 168-183 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] 坪川武弘: "工学系に必要とされる数学に関する調査研究1(材料力学)" 日本数学教育学会高専部会誌. 4・1. 99-128 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] 坪川武弘: "工学系に必要とされる数学に関する調査研究1(流体力学)" 日本数学教育学会高専部会誌. 4・1. 1-28 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Shin-ichi Shimada: "A solvable model for line interaction" RIMS Kokyuroku Kyoto univ.994. 168-183 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] 島田伸一: "A solvable model for line interaction" 京都大学数理解析研究所講究録. 994. 168-183 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 坪川武弘: "工学系に必要とされる数学に関する調査研究1(材料力学)" 日本数学教育学会高専部会誌. 4・1. 99-128 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 坪川武弘: "工学系に必要とされる数学に関する調査研究1(流体力学)" 日本数学教育学会高専部会誌. 4・1. 1-28 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 島田伸一: "A solvable model for line interaction" 京都大学数理解析研究所講究録. (予定). (1997)

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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