Spectral gaps of the Schrödinger operators with periodic d'-interactions and Diophantine approximations

• Autores: Kazushi Yoshitomi
• Localización: Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, Vol. 143, Nº 1, 2007, págs. 185-199
• Idioma: inglés
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• Resumen

  • We study the spectral gaps of the Schrödinger operator where ?(0,2p) and are parameters. Let t=2p-?. Suppose that the ratio ?0:=t/? 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?8 and the Diophantine properties of ?0. In particular, we show that if ß1+ß2=0, then where M(?0) stands for the Markov constant of ?0