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

Published online by Cambridge University Press:  01 July 2007

KAZUSHI YOSHITOMI
KAZUSHI YOSHITOMI*
Affiliation:
Department of Mathematics, Tokyo Metropolitan UniversityMinami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan. e-mail: yositomi@comp.metro-u.ac.jp
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Abstract

We study the spectral gaps of the Schrödinger operatorwhere κ∈(0,2π) and are parameters. Let τ=2π−κ. Suppose that the ratio κ0:=τ/κ is irrational. We denote the jth gap of the spectrum of H by Gj, its length by |Gj|. We obtain a relationship between the asymptotic behaviour of |Gj| as j→∞ and the Diophantine properties of κ0. In particular, we show that if β1+β2=0, thenwhere M(κ0) stands for the Markov constant of κ0.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 185 - 199
Copyright
Copyright © Cambridge Philosophical Society 2007

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