Spectral asymptotics of periodic discrete Schrödinger operators,I

Abstract

Asymptotics, for coupling constant A tending to zero and to infinity, for the bands and gaps of the Schrödinger operator (H_Ag)(n)=g(n+1)+Ab_ng(n)+g(n−1) on l²(Z) with periodic potential b are studied. In this context special attention is paid to ‘asymptotic spectral powers’. Three different perturbation techniques are used to obtain results. In this first paper the emphasis lies on results that can be obtained by a first-order degenerate or arbitrary-order non-degenerate Rayleigh–Schrödinger perturbation calculus.