| Project/Area Number |
08640248
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | SETSUNAN UNIVERSITY (1997)
Fukui National College of Technology (1996) |
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Principal Investigator |
SHIMADA Shin-ich Setsunan University, Depertment of Mathematics, Associate Professor, 工学部, 助教授 (40196481)
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| 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)
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| Project Period (FY) |
1996 – 1997
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| Project Status |
Completed (Fiscal Year 1997)
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| 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)
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| Keywords | Schrodinger Operator / Scattering / Wave Operator / Wave Function / Interference / Scattering Amplitude / Schrodinger作用素 / 散乱 / 自己共役拡張 |
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| 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.
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