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

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) (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>.