The Structure of the Spectrum of Fourth-Order Differential Operators with Random Coefficients

In this paper it is proved that under certain conditions on the coefficients the random operator H = $H = frac{1}{{rleft( {x_t } right)}}left[ {frac{{d^2 }}{{dt^2 }}left( {frac{1}{{pleft( {x_t } right)}}frac{{d^2 }}{{dt^2 }}} right) + qleft( {x_t } right)} right], in R^1$ being a stationary, ergodic Markov process with compact State space K, has almost surely pure point spectrum and exponentially decreasing eigenfunctions. The method used here can be extended to operators corresponding to certain matrix Sturm-Liouville problems.