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>.

In this project I analyzed the spectrum of the Schrodinger operator with periodicδ'-interaction of the FormH = d ^ 2 / dx ^ 2 + > < Σ (_1 ∈ z) (beta delta '(x - 2 PI kappa - predominate 1) ten gamma delta' (1) x - 2 PI) in L ^ 2 RHere, kappa ∈ predominate PI (0, 2) and beta, gamma ∈ R - {0} are the 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>.

项目成果

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

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 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

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 143
• 作者: Ryuichi;Ashino;Akira Morimoto;Akira Morimoto;芦野隆一;Kazushi Yoshitomi
• 通讯作者: Kazushi Yoshitomi