Zariski closure of subgroups of the symplectic group and Lyapunov exponents of the Schrödinger operator on the strip

Abstract

We consider the Schrödinger equation with a random potential $$ - y_{n + 1} + (Q_n - EI)y_n - y_{n - 1} = 0,$$ whereQ n is a sequence of independent identically distributed random symmetricm×m-matrices with real valued elements,y n ∈ ℝm, −∞<n<∞,E is the real parameter, andI is the identity matrix. We show that if the smallest Jordan algebra of matrices containing the support of the distribution of matricesQ n coincides with Jordan algebra of all (real-valued) symmetric matrices then for all but (maybe) a finite number of values ofE all the Lyapunov exponents of our Schrödinger equation are different (and thus the spectrum of the corresponding Schrödinger operator is pure point).