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\noindent In [1] L. Polterovich and Z. Rudnick considered the behavior of aone-parameter subgroup of a Lie group under the influence of a sequence ofkicks. Among others they raise the following problem: {\it is the horocycleflow stably quasi-mixing on $SL(2,\mathbb{R})/\Gamma$?} Equivalently it can bereformulated in terms of boundedness of the sequences of products $ P_n(t) =\Phi_n H(t)\Phi_{n-1} H(t) \, ... \, \Phi_1 H(t) $ where $H(t) =\begin{pmatrix} 1 & t  0 & 1 \end{pmatrix}$ and $\Phi=\{\Phi_n\} \subset SL(2,\mathbb{R})$. We solvethis problem positively and as a consequence obtain the following applicationto the discrete Schr\"odinger equation \begin{equation*} q_{k+1} - (2+tc_k)q_k+ q_{k-1}=0, \qquad k\geq 1: \end{equation*} the set of values of the parameter$t$ for which the equation has only bounded solutions, has finite measure.

Growth of matrix products and mixing properties of the horocycle flow