Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math-ph/0005017

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

Vadim Kostrykin·Robert Schrader
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Functorial Theory of Defects in Abelian Chern-Simons Theory

Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.

math-ph↗
arXiv2026

Gradient nature of Laplacian growth

For a class of growth processes of Laplacian type in the plane, we suggest an interpretation as a ``gradient descent'' in the space of smooth closed curves. More precisely, we show that boundary of a growing domain moves along a gradient of a certain functional in the space of curves. In the simplest cases this functional is $\log (1/r)$, where $r$ is the external conformal radius of the growing domain.

math-ph↗
arXiv2026

Entanglement-Inducing Quantum Markov Processes

We introduce a new model for a system of interacting bosons placed in an array of sites. At its core is a nonlinear, nonlocal evolution equation, which we have dubbed the Schrödinger-Dirichlet equation. The construction is closely related to the Bose-Hubbard model and to a specific type of generalized bosons. In contrast to conventional mean-field closures, the resulting nonlinear dynamics need not preserve product structure and can generate entanglement from initially separable states. The relevant methods of analysis are based on harmonic analysis for the multiplicative group of positive rationals.

math-ph↗