Localization of eigenfunctions at zero for certain almost periodic operators

For every irrational number a! satisfying the property lim(n -->infinity) \ sin pi alpha n \(-1/n) = 1 and for every number beta > 1, it is shown that the difference equation xi+1 + xi(n-1) + 2 beta cos(2 pi alpha n+theta)xi(n) = 0, is an element of Z has a non-trivial solution {xi(n)} satisfying (lim) over bar(\ n \-->infini ty) \xi(n)\(1/\ n \) less than or equal to \beta \(-1) if and only if theta = 2 pi alpha n + 2 pi k +/- pi/2 for some n, k is an element of Z.