On the spectrum of the one-dimensional Schitdinger operators with periodic point -interactions

具有周期性点相互作用的一维席丁格算子的谱

• 批准号: 18540190
• 负责人: YOSHITOMI Kazushi
• 金额: $ 2.48万
• 依托单位: Tokyo Metropolitan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540190/
• 关键词: Point-interactions; Periodic potentials; SchrOdinger operators; Spectral gaps; Asymptotic behavior; Diophantine approximation

In this project I analyzed the spectrum of the Schrodinger operator with periodicδ'-interaction of the formH=-d^2/dx^<2+>Σ (_1∈z) (βδ' (x-κ-2π1) 十γδ' (x-2π1) ) in L^2 RHere, 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>.

在这个项目中，我分析了schrodinger操作员的光谱，其中formh = -d^2/dx^<2+>σ（_1∈Z）（l^2 rhere中）在l^2 rhere中，κ布（0,2π）和β，γ，γ-γr- r- {0} semaim and anc anc and anc anc and anc and anc and anc anc and和在原点支持的功能。根据H和Floquet-Bloch理论的电势的周期性，用σ（H）表示的H的光谱具有频带结构。令G代表σ（h）的jth空隙。我们将-2τ=2π -κ_0 =τ/κ。这项研究的主要结果给出了参数κ_0的长度的渐近行为与数量理论特性之间的关系。为了简要了解，我们介绍了一个数字理论对象。假设κ_0是非理性的。令M（κ_0）代表κ_____________________________的马尔可夫常数= sup {m> 0;存在无限多对（q，p）∈Z×n，因此q |qκ__0-p | <1/ml。该常数代表通过有理数的κ__的近似性。以下含义说明了相对关系。 IFβ+γ= 0，然后lim inf（_j→∞）| g_1 | =2π^2（κτM（κ__0））^<-1>。

项目成果

Dirac operators with periodic σ -interactions -spectral gaps and inhomogeneous Diophantine approximation

具有周期性 σ 相互作用谱间隙和非齐次丢番图近似的狄拉克算子

• 发表时间: 2007
• 作者: Kazushi;Yoshitomi
• 通讯作者: Yoshitomi

Spectral gaps of the Schrodinger operators with periodic δ,-interactions and Diophantine approximations

具有周期性 δ,-相互作用和丢番图近似的薛定谔算子的谱间隙

• 发表时间: 2007
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society (掲載受理済み)
• 作者: Kazushi;Yoshitomi;Kazushi Yoshitomi;KazushiYoshitom
• 通讯作者: KazushiYoshitom

Spectral gaps of the SchrOdinger operators with periodic σ'-interactions and Diophantine approximations

具有周期性 σ 相互作用和丢番图近似的薛定谔算子的谱间隙

• 发表时间: 2007
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society 143
• 作者: Kazushi;Yoshitomi
• 通讯作者: Yoshitomi

Periodic point interactions and Diophantine approximation

周期性点相互作用和丢番图近似

• 发表时间: 2006
• 作者: Kazushi;Yoshitomi
• 通讯作者: Yoshitomi

周期的な点相互作用とディオファントス近似

周期性点相互作用和丢番图近似

• 发表时间: 2006
• 作者: Kazushi;Yoshitomi;吉冨 和志;吉冨 和志
• 通讯作者: 吉冨 和志