On the spectrum of the one-dimensional Schitdinger operators with periodic point -interactions
| Project/Area Number |
18540190
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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 |
Basic analysis
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
YOSHITOMI Kazushi Tokyo Metropolitan University, Graduate School of Science and Engineering, Associate Professor (40304729)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,880,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥480,000)
Fiscal Year 2007: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2006: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | Point-interactions / Periodic potentials / SchrOdinger operators / Spectral gaps / Asymptotic behavior / Diophantine approximation |
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| Research Abstract |
In this project I analyzed the spectrum of the Schrodinger operator with periodicδ'-interaction of the form
H=-d^2/dx^<2+>Σ (_1∈z) (βδ' (x-κ-2π1) 十γδ' (x-2π1) ) in L^2 R
Here, κ∈ (0,2π) and β,γ∈R-{0} are parameters, and δ' is the derivative of the Dirac delta function supported at the origin. By the periodicity of the potential of H and the Floquet-Bloch theory, the spectrum of H, denoted by σ (H), has the band structure. Let G stand for the jth gap of σ (H). We put -2τ=2π-κandκ_0 =τ/κ. The main result of this research gives a relationship between the asymptotic behavior of the length of and the number-theoretical properties of the parameter κ_0. In order to see that briefly, we introduce a number-theoretical object. Suppose that κ_0 is irrational. Let M (κ_0) stand for the Markov constant of κ_0 :
M (κ_0) =SuP{m>0 ; there exist infinitely many pairs (q,p)∈Z×N such that q|qκ_0-p|<1/ml.
This constant represents the approximability of κ_0 by rational numbers. The following implication illustrates the aforementioned relationship.
Theorem. Ifβ+γ=0,then lim inf (_j→∞)|G_1|=2π^2 (κτM (κ_0))^<-1>.
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Report
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Research Products
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