Spectral gaps of the Schrödinger operators with periodic δ′-interactions and Diophantine approximations

We study the spectral gaps of the Schrödinger operatorwhere κ∈(0,2π) and$\beta_{1},\beta_{2}\in{\mathbb R}\setminus\{0\}$are parameters. Let τ=2π−κ. Suppose that the ratio κ0:=τ/κ is irrational. We denote thejth gap of the spectrum ofHbyGj, its length by |Gj|. We obtain a relationship between the asymptotic behaviour of |Gj| asj→∞ and the Diophantine properties of κ0. In particular, we show that if β1+β2=0, thenwhereM(κ0) stands for the Markov constant of κ0.