Perturbation theory for Lyapunov exponents of an Anderson model on a strip

It is proven that the inverse localization length of an Anderson model on a strip of width L is bounded above by L/λ2 for small values of the coupling constant λ of the disordered potential. For this purpose, a formalism is developed in order to calculate the bottom Lyapunov exponent associated with random products of large symplectic matrices perturbatively in the coupling constant of the randomness. 1 Main result and discussion The Anderson model describes electronic waves scattered at random obstacles. Here the physical space is supposed to be quasi one dimensional and given by an infinite strip of finite but large width L. Hence the Hilbert space is l(Z,C) and states therein will be decomposed as ψ = (ψ(n))n∈Z with ψ(n) ∈ C. The Anderson Hamiltonian on a strip is then defined by (HL(λ)ψ)(n) = −ψ(n + 1)− ψ(n− 1) + ∆Lψ(n) + λV (n)ψ(n) . Here ∆L : C L → C is the transverse (one dimensional) discrete laplacian with periodic boundary conditions; denoting the cyclic shift on C by S, it is given by ∆L = −S − S. For L = 1, 2, one may rather set ∆1 = 0 and ∆2 = −S = −S∗, but our main interest will be in the case L ≥ 3 anyway. Furthermore, λ ∈ R is the coupling constant of the random potential V (n) : C → C which is a diagonal matrix V (n) = diag(v(n, 1), . . . , v(n, L)). All real numbers (v(n, l))n∈Z, l=1,...,L are independent and identically distributed centered real variables with unit variance. Given a fixed energy E ∈ R, it is convenient to rewrite the eigenvalue equation HL(λ)ψ = Eψ using the transfer matrices ( ψ(n+ 1) ψ(n) ) = T (n) ( ψ(n) ψ(n− 1) ) , T (n) = ( ∆L + λV (n)− E 1 −1 1 0 ) . (1)