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Global Bounds for the Lyapunov Exponent and the Integrated Density of States of Random Schrödinger Operators in One Dimension

Vadim Kostrykin·Robert Schrader
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Abstract

In this article we prove an upper bound for the Lyapunov exponent $γ(E)$ and a two-sided bound for the integrated density of states $N(E)$ at an arbitrary energy $E>0$ of random Schrödinger operators in one dimension. These Schrödinger operators are given by potentials of identical shape centered at every lattice site but with non-overlapping supports and with randomly varying coupling constants. Both types of bounds only involve scattering data for the single-site potential. They show in particular that both $γ(E)$ and $N(E)-\sqrt{E}/π$ decay at infinity at least like $1/\sqrt{E}$. As an example we consider the random Kronig-Penney model.

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Vadim Kostrykin, Robert Schrader. 2000-05-15. Global Bounds for the Lyapunov Exponent and the Integrated Density of States of Random Schrödinger Operators in One Dimension. https://doi.org/10.1088/0305-4470%2F33%2F46%2F306

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