 00226 Vadim Kostrykin and Robert Schrader
 Global Bounds for the Lyapunov Exponent and the Integrated Density
of States of Random Schr\"odinger Operators in One Dimension
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May 15, 00

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Abstract. In this article we prove an upper bound for the Lyapunov exponent $\gamma(E)$ and a twosided bound for the integrated density of states $N(E)$ at an arbitrary energy $E>0$ of random Schr\"odinger operators in one dimension. These Schr\"odinger operators are given by potentials of identical shape centered at every lattice site but with nonoverlapping supports and with randomly varying coupling constants. Both types of bounds only involve scattering data for the singlesite potential. They show in particular that both $\gamma(E)$ and
$N(E)\sqrt{E}/\pi$ decay at infinity at least like $1/\sqrt{E}$. As an example we consider the random KronigPenney model.
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