Dynamics of a Disordered Linear Chain

By a disordered linear chain, we mean a chain of one-dimensional harmonic linear oscillators, each coupled to its nearest neighbors by harmonic forces, with the mass of each oscillator and the coupling parameters taken to be random variables with known distributions. The problem of calculating the distributions. The problem of calculating the distribution function of the frequencies of the normal modes of vibration of the chain in the limit as the chain becomes infinitely long was resolved by Dyson. In this paper, we present a simple algebraic proof of the essential limit relation in Dyson's paper.