Large coupling behaviour of the Lyaponov exponent for tight binding one - dimensional random systems ?

We study the Lyaponov exponent y A (E) of (h u) (n) = u(n + 1) + u(n-l) + A V (n) u (n) in the limit as A +a where V is a suitable random potential. We prove that yA(E)-ln A as A +CO uniformly as E/A runs through compact sets. We also describe a formal expansion (to order A-*) for random and almost periodic potentials. In this note, we study one-dimensional tight binding Hamiltonians h = ho + A V where (1) We are interested in the cases where V is either random or almost periodic. By random, we mean that V (n) is a family of identically distributed independent random variables with density P(y)dy where P is bounded with bounded support. In the random case, we will succeed in identifying the first few terms in the large A behaviour of the Lyaponov exponent. For the almost periodic case only a formal large A expansion is obtained. (hou)(n) = u (n + 1) + u (n-1). Explicitly, we let yA (E) be the Lyaponov exponent for h, i.e. 1 yA(E)= lim-lnllM,,(u).. .Ml(w)ll n+m n where The limit exists for typical U. We will prove that in the random case as A + CO, for E fixed, YA(AE)-ln A-lnlE-E'IP(E') dE'+O. J As we will explain, we believe but will not prove that the left side of (2) is actually O(A-'). A formal calculation of the A-' term will be given subsequently.