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

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

2000-05-15

arxiv.org/abs/math-ph/0005017v1

J. Phys. A: Math. Gen. 33 (2000) 8231 - 8240

Physics

Mathematical Physics

9 pages

Scientific paper

10.1088/0305-4470/33/46/306

In this article we prove an upper bound for the Lyapunov exponent $\gamma(E)$ and a two-sided 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 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 $\gamma(E)$ and $N(E)-\sqrt{E}/\pi$ decay at infinity at least like $1/\sqrt{E}$. As an example we consider the random Kronig-Penney model.

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